Keywords
 A2.1.11. Proof languages
 A2.4.3. Proofs
 A4.5. Formal methods for security
 A7.2. Logic in Computer Science
 A8.1. Discrete mathematics, combinatorics
 A8.3. Geometry, Topology
 A8.4. Computer Algebra
 A8.5. Number theory
 B9.5.2. Mathematics
 B9.5.3. Physics
1 Team members, visitors, external collaborators
Research Scientists
 Frédéric Chyzak [Team leader, Inria, Researcher, HDR]
 Alin Bostan [Inria, Researcher, HDR]
 Guy Fayolle [Inria, Emeritus]
 Georges Gonthier [Inria, Senior Researcher, until Oct 2020]
 Pierre Lairez [Inria, Researcher]
Faculty Member
 Philippe Dumas [Ministère de l'Education Nationale]
PhD Students
 Alexandre Goyer [Inria, from Oct 2020]
 Rafael Mohr [Technische Universität Kaiserslautern (Germany), from Nov 2020]
 Raphaël Pagès [Université de Bordeaux, from Sep 2020]
 Sergey Yurkevich [University of Vienna (Austria), from Oct 2020]
Interns and Apprentices
 Alexandre Goyer [Sorbonne Université, from Mar 2020 until Jul 2020]
 Lucas Morisset [Inria, from Jun 2020 until Jul 2020]
 Raphaël Pagès [Université Paris Diderot, from Mar 2020 until Jul 2020]
 Nicholas Rumiz [Inria, from Mar 2020 until Jul 2020]
Administrative Assistant
 Bahar Carabetta [Inria]
Visiting Scientist
 Antonio Jiménez Pastor [Johannes Kepler University (Linz, Austria), from Feb 2020 until May 2020]
External Collaborator
 Marc Mezzarobba [CNRS]
2 Overall objectives
2.1 Scientific challenges, expected impact
The general orientation of our team is described by the short name given to it: Special Functions, that is, particular mathematical functions that have established names due to their importance in mathematical analysis, physics, and other application domains. Indeed, we ambition to study special functions with the computer, by combined means of computer algebra and formal methods.
Computeralgebra systems have been advertised for decades as software for “doing mathematics by computer” 86. For instance, computeralgebra libraries can uniformly generate a corpus of mathematical properties about special functions, so as to display them on an interactive website. This possibility was recently shown by the computeralgebra component of the team 40. Such an automated generation significantly increases the reliability of the mathematical corpus, in comparison to the content of existing static authoritative handbooks. The importance of the validity of these contents can be measured by the very wide audience that such handbooks have had, to the point that a book like 37 remains one of the most cited mathematical publications ever and has motivated the 10yearlong project of writing its successor 76. However, can the mathematics produced “by computer” be considered as true mathematics? More specifically, whereas it is nowadays well established that the computer helps in discovering and observing new mathematical phenomenons, can the mathematical statements produced with the aid of the computer and the mathematical results computed by it be accepted as valid mathematics, that is, as having the status of mathematical proofs? Beyond the reported weaknesses or controversial design choices of mainstream computeralgebra systems, the issue is more of an epistemological nature. It will not find its solution even in the advent of the ultimate computeralgebra system: the social process of peerreviewing just falls short of evaluating the results produced by computers, as reported by Th. Hales 64 after the publication of his proof of the Kepler Conjecture about sphere packing.
A natural answer to this deadlock is to move to an alternative kind of mathematical software and to use a proof assistant to check the correctness of the desired properties or formulas. The success of largescale formalization projects, like the FourColor Theorem of graph theory 59, the abovementioned Kepler Conjecture 64, and the Odd Order Theorem of group theory 1, have increased the understanding of the appropriate softwareengineering methods for this peculiar kind of programming. For computer algebra, this legitimates a move to proof assistants now.
The Dynamic Dictionary of Mathematical Functions 2 (DDMF) 40 is an online computergenerated handbook of mathematical functions that ambitions to serve as a reference for a broad range of applications. This software was developed by the computeralgebra component of the team as a project 3 of the MSR–Inria Joint Centre. It bases on a library for the computeralgebra system Maple, Algolib 4, whose development started 20 years ago in projectteam Algorithms 5. As suggested by the constant questioning of certainty by new potential users, DDMF deserves a formal guarantee of correctness of its content, on a level that proof assistants can provide. Fortunately, the maturity of specialfunctions algorithms in Algolib makes DDMF a stepping stone for such a formalization: it provides a wellunderstood and unified algorithmic treatment, without which a formal certification would simply be unreachable.
The formalproofs component of the team emanates from another project of the MSR–Inria Joint Centre, namely the Mathematical Components project (MathComp) 6. Since 2006, the MathComp group has endeavoured to develop computerchecked libraries of formalized mathematics, using the Coq proof assistant 83. The methodological aim of the project was to understand the design methods leading to successful largescale formalizations. The work culminated in 2012 with the completion of a formal proof of the Odd Order Theorem, resulting in the largest corpus of algebraic theories ever machinechecked with a proof assistant and a whole methodology to effectively combine these components in order to tackle complex formalizations. In particular, these libraries provide a good number of the many algebraic objects needed to reason about special functions and their properties, like rational numbers, iterated sums, polynomials, and a rich hierarchy of algebraic structures.
The present team takes benefit from these recent advances to explore the formal certification of the results collected in DDMF. The aim of this project is to concentrate the formalization effort on this delimited area, building on DDMF and the Algolib library, as well as on the Coq system 83 and on the libraries developed by the MathComp project.
2.2 Use computer algebra but convince users beyond reasonable doubt
The following few opinions on computer algebra are, we believe, typical of computeralgebra users' doubts and difficulties when using computeralgebra systems:
 Fredrik Johansson, expert in the multiprecision numerical evaluation of special functions and in fast computeralgebra algorithms, writes on his blog 70: “Mathematica is great for crosschecking numerical values, but it's not unusual to run into bugs, so triple checking is a good habit.” One answer in the discussion is: “We can claim that Mathematica has [...] an impossible to understand semantics: If Mathematica's output is wrong then change the input. If you don't like the answer, change the question. That seems to be the philosophy behind.”
 A professor's advice to students 78 on using Maple: “You may wish to use Maple to check your homework answers. If you do then keep in mind that Maple sometimes gives the wrong answer, usually because you asked incorrectly, or because of niceties of analytic continuation. You may even be bitten by an occasional Maple bug, though that has become fairly unlikely. Even with as powerful a tool as Maple you will still have to devise your own checks and you will still have to think.”
 Jacques Carette, former head of the maths group at Maplesoft, about a bug 53 when asking Maple to take the limit limit(f(n) * exp(n), n = infinity) for an undetermined function f: “The problem is that there is an implicit assumption in the implementation that unknown functions do not `grow too fast'.”
As explained by the expert views above, complaints by computeralgebra users are often due to their misunderstanding of what a computeralgebra systems is, namely a purely syntactic tool for calculations, that the user must complement with a semantics. Still, robustness and consistency of computeralgebra systems are not ensured as of today, and, whatever Zeilberger may provocatively say in his Opinion 94 88, a firmer logical foundation is necessary. Indeed, the fact is that many “bugs” in a computeralgebra system cannot be fixed by just the usual debugging method of tracking down the faulty lines in the code. It is sort of “by design”: assumptions that too often remain implicit are really needed by the design of symbolic algorithms and cannot easily be expressed in the programming languages used in computer algebra. A similar certification initiative has already been undertaken in the domain of numerical computing, in a successful manner 66, 43. It is natural to undertake a similar approach for computer algebra.
2.3 Make computer algebra and formal proofs help one another
Some of the mathematical objects that interest our team are still totally untouched by formalization. When implementing them and their theory inside a proof assistant, we have to deal with the pervasive discrepancy between the published literature and the actual implementation of computeralgebra algorithms. Interestingly, this forces us to clarify our computeralgebraic view on them, and possibly make us discover holes lurking in published (human) proofs. We are therefore convinced that the close interaction of researchers from both fields, which is what we strive to maintain in this team, is a strong asset.
For a concrete example, the core of Zeilberger's creative telescoping manipulates rational functions up to simplifications. In summation applications, checking that these simplifications do not hide problematic divisions by 0 is most often left to the reader. In the same vein, in the case of integrals, the published algorithms do not check the convergence of all integrals, especially in intermediate calculations. Such checks are again left to the readers. In general, we expect to revisit the existing algorithms to ensure that they are meaningful for genuine mathematical sequences or functions, and not only for algebraic idealizations.
Another big challenge in this project originates in the scientific difference between computer algebra and formal proofs. Computer algebra seeks speed of calculation on concrete instances of algebraic data structures (polynomials, matrices, etc). For their part, formal proofs manipulate symbolic expressions in terms of abstract variables understood to represent generic elements of algebraic data structures. In view of this, a continuous challenge is to develop the right, hybrid thinking attitude that is able to effectively manage concrete and abstract values simultaneously, alternatively computing and proving with them.
2.4 Experimental mathematics with special functions
Applications in combinatorics and mathematical physics frequently involve equations of so high orders and so large sizes, that computing or even storing all their coefficients is impossible on existing computers. Making this tractable is an extraordinary challenge. The approach we believe in is to design algorithms of good—ideally quasioptimal—complexity in order to extract precisely the required data from the equations, while avoiding the computationally intractable task of completely expanding them into an explicit representation.
Typical applications with expected high impact are the automatic discovery and algorithmic proof of results in combinatorics and mathematical physics for which human proofs are currently unattainable.
2.5 Research axes
The implementation of certified symbolic computations on special functions in the Coq proof assistant requires both investigating new formalization techniques and renewing the traditional computeralgebra viewpoint on these standard objects. Large mathematical objects typical of computer algebra occur during formalization, which also requires us to improve the efficiency and ergonomics of Coq. In order to feed this interdisciplinary activity with new motivating problems, we additionally pursue a research activity oriented towards experimental mathematics in application domains that involve special functions. We expect these applications to pose new algorithmic challenges to computer algebra, which in turn will deserve a formalcertification effort. Finally, DDMF is the motivation and the showcase of our progress on the certification of these computations. While striving to provide a formal guarantee of the correctness of the information it displays, we remain keen on enriching its mathematical content by developing new computeralgebra algorithms.
2.6 Computer algebra certified by the Coq system
Our formalization effort consists in organizing a cooperation between a computeralgebra system and a proof assistant. The computeralgebra system is used to produce efficiently algebraic data, which are later processed by the proof assistant. The success of this cooperation relies on the design of appropriate libraries of formalized mathematics, including certified implementations of certain computeralgebra algorithms. On the other side, we expect that scrutinizing the implementation and the output of computeralgebra algorithms will shed a new light on their semantics and on their correctness proofs, and help clarifying their documentation.
2.6.1 Libraries of formalized mathematics
The appropriate framework for the study of efficient algorithms for special functions is algebraic. Representing algebraic theories as Coq formal libraries takes benefit from the methodology emerging from the success of ambitious projects like the formal proof of a major classification result in finitegroup theory (the Odd Order Theorem) 58.
Yet, a number of the objects we need to formalize in the present context has never been investigated using any interactive proof assistant, despite being considered as commonplaces in computer algebra. For instance there is up to our knowledge no available formalization of the theory of noncommutative rings, of the algorithmic theory of specialfunctions closures, or of the asymptotic study of special functions. We expect our future formal libraries to prove broadly reusable in later formalizations of seemingly unrelated theories.
2.6.2 Manipulation of large algebraic data in a proof assistant
Another peculiarity of the mathematical objects we are going to manipulate with the Coq system is their size. In order to provide a formal guarantee on the data displayed by DDMF, two related axes of research have to be pursued. First, efficient algorithms dealing with these large objects have to be programmed and run in Coq. Recent evolutions of the Coq system to improve the efficiency of its internal computations 38, 41 make this objective reachable. Still, how to combine the aforementioned formalization methodology with these cuttingedge evolutions of Coq remains one of the prospective aspects of our project. A second need is to help users interactively manipulate large expressions occurring in their conjectures, an objective for which little has been done so far. To address this need, we work on improving the ergonomics of the system in two ways: first, ameliorating the reactivity of Coq in its interaction with the user; second, designing and implementing extensions of its interface to ease our formalization activity. We expect the outcome of these lines of research to be useful to a wider audience, interested in manipulating large formulas on topics possibly unrelated to special functions.
2.6.3 Formalproofproducing normalization algorithms
Our algorithm certifications inside Coq intend to simulate wellidentified components of our Maple packages, possibly by reproducing them in Coq. It would however not have been judicious to reimplement them inside Coq in a systematic way. Indeed for a number of its components, the output of the algorithm is more easily checked than found, like for instance the solving of a linear system. Rather, we delegate the discovery of the solutions to an external, untrusted oracle like Maple. Trusted computations inside Coq then formally validate the correctness of the a priori untrusted output. More often than not, this validation consists in implementing and executing normalization procedures inside Coq. A challenge of this automation is to make sure they go to scale while remaining efficient, which requires a Coq version of nontrivial computeralgebra algorithms. A first, archetypal example we expect to work on is a noncommutative generalization of the normalization procedure for elements of rings 63.
2.7 Better symbolic computations with special functions
Generally speaking, we design algorithms for manipulating special functions symbolically, whether univariate or with parameters, and for extracting algorithmically any kind of algebraic and analytic information from them, notably asymptotic properties. Beyond this, the heart of our research is concerned with parametrised definite summations and integrations. These very expressive operations have farranging applications, for instance, to the computation of integral transforms (Laplace, Fourier) or to the solution of combinatorial problems expressed via integrals (coefficient extractions, diagonals). The algorithms that we design for them need to really operate on the level of linear functional systems, differential and of recurrence. In all cases, we strive to design our algorithms with the constant goal of good theoretical complexity, and we observe that our algorithms are also fast in practice.
2.7.1 Specialfunction integration and summation
Our longterm goal is to design fast algorithms for a general method for specialfunction integration (creative telescoping), and make them applicable to general specialfunction inputs. Still, our strategy is to proceed with simpler, more specific classes first (rational functions, then algebraic functions, hyperexponential functions, Dfinite functions, nonDfinite functions; two variables, then many variables); as well, we isolate analytic questions by first considering types of integration with a more purely algebraic flavor (constant terms, algebraic residues, diagonals of combinatorics). In particular, we expect to extend our recent approach 46 to more general classes (algebraic with nested radicals, for example): the idea is to speed up calculations by making use of an analogue of Hermite reduction that avoids considering certificates. Homologous problems for summation will be addressed as well.
2.7.2 Applications to experimental mathematics
As a consequence of our complexitydriven approach to algorithms design, the algorithms mentioned in the previous paragraph are of good complexity. Therefore, they naturally help us deal with applications that involve equations of high orders and large sizes.
With regard to combinatorics, we expect to advance the algorithmic classification of combinatorial classes like walks and urns. Here, the goal is to determine if enumerative generating functions are rational, algebraic, or Dfinite, for example. Physical problems whose modelling involves specialfunction integrals comprise the study of models of statistical mechanics, like the Ising model for ferromagnetism, or questions related to Hamiltonian systems.
Number theory is another promising domain of applications. Here, we attempt an experimental approach to the automated certification of integrality of the coefficients of mirror maps for Calabi–Yau manifolds. This could also involve the discovery of new Calabi–Yau operators and the certification of the existing ones. We also plan to algorithmically discover and certify new recurrences yielding good approximants needed in irrationality proofs.
It is to be noted that in all of these application domains, we would so far use general algorithms, as was done in earlier works of ours 45, 49, 48. To push the scale of applications further, we plan to consider in each case the specifics of the application domain to tailor our algorithms.
2.8 Interactive and certified mathematical web sites
In continuation of our past project of an encyclopedia at
http://
 the algorithmic discussion of equations with parameters, leading to certified automatic case analysis based on arithmetic properties of the parameters;
 lists of summation and integral formulas involving special functions, including validity conditions on the parameters;
 guaranteed largeprecision numerical evaluations.
3 Research program
3.1 Studying special functions by computer algebra
Computer algebra manipulates symbolic representations of exact mathematical objects in a computer, in order to perform computations and operations like simplifying expressions and solving equations for “closedform expressions”. The manipulations are often fundamentally of algebraic nature, even when the ultimate goal is analytic. The issue of efficiency is a particular one in computer algebra, owing to the extreme swell of the intermediate values during calculations.
Our view on the domain is that research on the algorithmic manipulation of special functions is anchored between two paradigms:
 adopting linear differential equations as the right data structure for special functions,
 designing efficient algorithms in a complexitydriven way.
It aims at four kinds of algorithmic goals:
 algorithms combining functions,
 functional equations solving,
 multiprecision numerical evaluations,
 guessing heuristics.
This interacts with three domains of research:
 computer algebra, meant as the search for quasioptimal algorithms for exact algebraic objects,
 symbolic analysis/algebraic analysis;
 experimental mathematics (combinatorics, mathematical physics, ...).
This view is made explicit in the present section.
3.2 Equations as a data structure
Numerous special functions satisfy linear differential and/or recurrence equations. Under a mild technical condition, the existence of such equations induces a finiteness property that makes the main properties of the functions decidable. We thus speak of Dfinite functions. For example, 60 % of the chapters in the handbook 37 describe Dfinite functions. In addition, the class is closed under a rich set of algebraic operations. This makes linear functional equations just the right data structure to encode and manipulate special functions. The power of this representation was observed in the early 1990s 87, leading to the design of many algorithms in computer algebra. Both on the theoretical and algorithmic sides, the study of Dfinite functions shares much with neighbouring mathematical domains: differential algebra, Dmodule theory, differential Galois theory, as well as their counterparts for recurrence equations.
3.3 Algorithms combining functions
Differential/recurrence equations that define special functions can be recombined 87 to define: additions and products of special functions; compositions of special functions; integrals and sums involving special functions. Zeilberger's fast algorithm for obtaining recurrences satisfied by parametrised binomial sums was developed in the early 1990s already 89. It is the basis of all modern definite summation and integration algorithms. The theory was made fully rigorous and algorithmic in later works, mostly by a group in Risc (Linz, Austria) and by members of the team 77, 85, 52, 50, 51, 71. The past ÉPI Algorithms contributed several implementations (gfun80, Mgfun52).
3.4 Solving functional equations
Encoding special functions as defining linear functional equations postpones some of the difficulty of the problems to a delayed solving of equations. But at the same time, solving (for special classes of functions) is a subtask of many algorithms on special functions, especially so when solving in terms of polynomial or rational functions. A lot of work has been done in this direction in the 1990s; more intensively since the 2000s, solving differential and recurrence equations in terms of special functions has also been investigated.
3.5 Multiprecision numerical evaluation
A major conceptual and algorithmic difference exists for numerical calculations between data structures that fit on a machine word and data structures of arbitrary length, that is, multiprecision arithmetic. When multiprecision floatingpoint numbers became available, early works on the evaluation of special functions were just promising that “most” digits in the output were correct, and performed by heuristically increasing precision during intermediate calculations, without intended rigour. The original theory has evolved in a twofold way since the 1990s: by making computable all constants hidden in asymptotic approximations, it became possible to guarantee a prescribed absolute precision; by employing stateoftheart algorithms on polynomials, matrices, etc, it became possible to have evaluation algorithms in a time complexity that is linear in the output size, with a constant that is not more than a few units. On the implementation side, several original works exist, one of which (NumGfun75) is used in our DDMF.
3.6 Guessing heuristics
“Differential approximation”, or “Guessing”, is an operation to get an ODE likely to be satisfied by a given approximate series expansion of an unknown function. This has been used at least since the 1970s and is a key stone in spectacular applications in experimental mathematics 49. All this is based on subtle algorithms for Hermite–Padé approximants 39. Moreover, guessing can at times be complemented by proven quantitative results that turn the heuristics into an algorithm 47. This is a promising algorithmic approach that deserves more attention than it has received so far.
3.7 Complexitydriven design of algorithms
The main concern of computer algebra has long been to prove the feasibility of a given problem, that is, to show the existence of an algorithmic solution for it. However, with the advent of faster and faster computers, complexity results have ceased to be of theoretical interest only. Nowadays, a large track of works in computer algebra is interested in developing fast algorithms, with time complexity as close as possible to linear in their output size. After most of the more pervasive objects like integers, polynomials, and matrices have been endowed with fast algorithms for the main operations on them 90, the community, including ourselves, started to turn its attention to differential and recurrence objects in the 2000s. The subject is still not as developed as in the commutative case, and a major challenge remains to understand the combinatorics behind summation and integration. On the methodological side, several paradigms occur repeatedly in fast algorithms: “divide and conquer” to balance calculations, “evaluation and interpolation” to avoid intermediate swell of data, etc. 44.
3.8 Encyclopedias
Handbooks collecting mathematical properties aim at serving as reference, therefore trusted, documents. The decision of several authors or maintainers of such knowledge bases to move from paper books 37, 76, 81 to websites and wikis 7 allows for a more collaborative effort in proof reading. Another step toward further confidence is to manage to generate the content of an encyclopedia by computeralgebra programs, as is the case with the Wolfram Functions Site 8 or DDMF 9. Yet, due to the lingering doubts about computeralgebra systems, some encyclopedias propose both crosschecking by different systems and handwritten companion paper proofs of their content 10. As of today, there is no encyclopedia certified with formal proofs.
3.9 Computer algebra and symbolic logic
Several attempts have been made in order to extend existing computeralgebra systems with symbolic manipulations of logical formulas. Yet, these works are more about extending the expressivity of computeralgebra systems than about improving the standards of correctness and semantics of the systems. Conversely, several projects have addressed the communication of a proof system with a computeralgebra system, resulting in an increased automation available in the proof system, to the price of the uncertainty of the computations performed by this oracle.
3.10 Certifying systems for computer algebra
More ambitious projects have tried to design a new computeralgebra system providing an environment where the user could both program efficiently and elaborate formal and machinechecked proofs of correctness, by calling a generalpurpose proof assistant like the Coq system. This approach requires a huge manpower and a daunting effort in order to reimplement a complete computeralgebra system, as well as the libraries of formal mathematics required by such formal proofs.
3.11 Semantics for computer algebra
The move to machinechecked proofs of the mathematical correctness of the output of computeralgebra implementations demands a prior clarification about the often implicit assumptions on which the presumably correctly implemented algorithms rely. Interestingly, this preliminary work, which could be considered as independent from a formal certification project, is seldom precise or even available in the literature.
3.12 Formal proofs for symbolic components of computeralgebra systems
A number of authors have investigated ways to organize the communication of a chosen computeralgebra system with a chosen proof assistant in order to certify specific components of the computeralgebra systems, experimenting various combinations of systems and various formats for mathematical exchanges. Another line of research consists in the implementation and certification of computeralgebra algorithms inside the logic 84, 63, 72 or as a proofautomation strategy. Normalization algorithms are of special interest when they allow to check results possibly obtained by an external computeralgebra oracle 56. A discussion about the systematic separation of the search for a solution and the checking of the solution is already clearly outlined in 69.
3.13 Formal proofs for numerical components of computeralgebra systems
Significant progress has been made in the certification of numerical applications by formal proofs. Libraries formalizing and implementing floatingpoint arithmetic as well as large numbers and arbitraryprecision arithmetic are available. These libraries are used to certify floatingpoint programs, implementations of mathematical functions and for applications like hybrid systems.
3.14 Machinechecked proofs of formalized mathematics
To be checked by a machine, a proof needs to be expressed in a constrained, relatively simple formal language. Proof assistants provide facilities to write proofs in such languages. But, as merely writing, even in a formal language, does not constitute a formal proof just per se, proof assistants also provide a proof checker: a small and wellunderstood piece of software in charge of verifying the correctness of arbitrarily large proofs. The gap between the lowlevel formal language a machine can check and the sophistication of an average page of mathematics is conspicuous and unavoidable. Proof assistants try to bridge this gap by offering facilities, like notations or automation, to support convenient formalization methodologies. Indeed, many aspects, from the logical foundation to the user interface, play an important role in the feasibility of formalized mathematics inside a proof assistant.
3.15 Logical foundations and proof assistants
While many logical foundations for mathematics have been proposed, studied, and implemented, type theory is the one that has been more successfully employed to formalize mathematics, to the notable exception of the Mizar system 73, which is based on set theory. In particular, the calculus of construction (CoC) 54 and its extension with inductive types (CIC) 55, have been studied for more than 20 years and been implemented by several independent tools (like Lego, Matita, and Agda). Its reference implementation, Coq 83, has been used for several largescale formalizations projects (formal certification of a compiler backend; fourcolor theorem). Improving the type theory underlying the Coq system remains an active area of research. Other systems based on different type theories do exist and, whilst being more oriented toward software verification, have been also used to verify results of mainstream mathematics (primenumber theorem; Kepler conjecture).
3.16 Computations in formal proofs
The most distinguishing feature of CoC is that computation is promoted to the status of rigorous logical argument. Moreover, in its extension CIC, we can recognize the key ingredients of a functional programming language like inductive types, pattern matching, and recursive functions. Indeed, one can program effectively inside tools based on CIC like Coq. This possibility has paved the way to many effective formalization techniques that were essential to the most impressive formalizations made in CIC.
Another milestone in the promotion of the computationsasproofs feature of Coq has been the integration of compilation techniques in the system to speed up evaluation. Coq can now run realistic programs in the logic, and hence easily incorporates calculations into proofs that demand heavy computational steps.
Because of their different choice for the underlying logic, other proof assistants have to simulate computations outside the formal system, and indeed fewer attempts to formalize mathematical proofs involving heavy calculations have been made in these tools. The only notable exception, which was finished in 2014, the Kepler conjecture, required a significant work to optimize the rewriting engine that simulates evaluation in Isabelle/HOL.
3.17 Largescale computations for proofs inside the Coq system
Programs run and proved correct inside the logic are especially useful for the conception of automated decision procedures. To this end, inductive types are used as an internal language for the description of mathematical objects by their syntax, thus enabling programs to reason and compute by case analysis and recursion on symbolic expressions.
The output of complex and optimized programs external to the proof assistant can also be stamped with a formal proof of correctness when their result is easier to check than to find. In that case one can benefit from their efficiency without compromising the level of confidence on their output at the price of writing and certify a checker inside the logic. This approach, which has been successfully used in various contexts, is very relevant to the present research project.
3.18 Relevant contributions from the Mathematical Component libraries
Representing abstract algebra in a proof assistant has been studied for long. The libraries developed by the MathComp project for the proof of the Odd Order Theorem provide a rather comprehensive hierarchy of structures; however, they originally feature a large number of instances of structures that they need to organize. On the methodological side, this hierarchy is an incarnation of an original work 58 based on various mechanisms, primarily type inference, typically employed in the area of programming languages. A large amount of information that is implicit in handwritten proofs, and that must become explicit at formalization time, can be systematically recovered following this methodology.
Smallscale reflection 60 is another methodology promoted by the MathComp project. Its ultimate goal is to ease formal proofs by systematically dealing with as many bureaucratic steps as possible, by automated computation. For instance, as opposed to the style advocated by Coq's standard library, decidable predicates are systematically represented using computable boolean functions: comparison on integers is expressed as program, and to state that $a\le b$ one compares the output of this program run on $a$ and $b$ with $true$. In many cases, for example when $a$ and $b$ are values, one can prove or disprove the inequality by pure computation.
The MathComp library was consistently designed after uniform principles of software engineering. These principles range from simple ones, like naming conventions, to more advanced ones, like generic programming, resulting in a robust and reusable collection of formal mathematical components. This large body of formalized mathematics covers a broad panel of algebraic theories, including of course advanced topics of finite group theory, but also linear algebra, commutative algebra, Galois theory, and representation theory. We refer the interested reader to the online documentation of these libraries 82, which represent about 150,000 lines of code and include roughly 4,000 definitions and 13,000 theorems.
Topics not addressed by these libraries and that might be relevant to the present project include real analysis and differential equations. The most advanced work of formalization on these domains is available in the HOLLight system 65, 67, 68, although some existing developments of interest 42, 74 are also available for Coq. Another aspect of the MathComp libraries that needs improvement, owing to the size of the data we manipulate, is the connection with efficient data structures and implementations, which only starts to be explored.
3.19 User interaction with the proof assistant
The user of a proof assistant describes the proof he wants to formalize in the system using a textual language. Depending on the peculiarities of the formal system and the applicative domain, different proof languages have been developed. Some proof assistants promote the use of a declarative language, when the Coq and Matita systems are more oriented toward a procedural style.
The development of the large, consistent body of MathComp libraries has prompted the need to design an alternative and coherent language extension for the Coq proof assistant 62, 61, enforcing the robustness of proof scripts to the numerous changes induced by code refactoring and enhancing the support for the methodology of smallscale reflection.
The development of large libraries is quite a novelty for the Coq system. In particular any longterm development process requires the iteration of many refactoring steps and very little support is provided by most proof assistants, with the notable exception of Mizar 79. For the Coq system, this is an active area of research.
4 Application domains
4.1 Computer Algebra in Mathematics
Our expertise in computer algebra and complexitydriven design of algebraic algorithms has applications in various domains, including:
 combinatorics, especially the study of combinatorial walks,
 theoretical computer science, like by the study of automatic sequences,
 number theory, by the analysis of the nature of socalled periods.
5 New results
5.1 Algebraic algorithms on fundamental objects
5.1.1 A simple and fast algorithm for computing the $N$th term of a linearly recurrent sequence
In 24 Alin Bostan and Ryuhei Mori (Tokyo Institute of Technology, Japan) designed a simple and fast algorithm for computing the $N$th term of a given linearly recurrent sequence. The new algorithm uses $O\left(\U0001d5ac\right(d)logN)$ arithmetic operations, where $d$ is the order of the recurrence, and $\U0001d5ac\left(d\right)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The stateoftheart algorithm, due to Fiduccia (1985), had the same arithmetic complexity up to a constant factor. The new algorithm is simpler, faster and obtained by a totally different method. They also discuss several algorithmic applications, notably to polynomial modular exponentiation (${P}^{N}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}Q$), on which many other useful algorithms rely, either in computer algebra (e.g., polynomial factoring over finite fields), or in algorithmic number theory (e.g., primality tests) or in effective algebraic geometry (e.g., counting points on curves over finite fields).
5.1.2 Fast computation of the $N$th term of a $q$holonomic sequence and applications
In 1977, Strassen invented a famous babystep/giantstep algorithm that computes the factorial $N!$ in arithmetic complexity quasilinear in $\sqrt{N}$. In 1988, the Chudnovsky brothers generalized Strassen’s algorithm to the computation of the $N$th term of any holonomic sequence in essentially the same arithmetic complexity. In 23, 29, Alin Bostan together with his PhD student Sergey Yurkevich designed $q$analogues of these algorithms. They first extend Strassen’s algorithm to the computation of the $q$factorial of $N$, then Chudnovskys' algorithm to the computation of the $N$th term of any $q$holonomic sequence. Both algorithms work in arithmetic complexity quasilinear in $\sqrt{N}$; surprisingly, they are simpler than their analogues in the holonomic case. They provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, they describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear $q$differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.
5.1.3 Subresultants of ${(x\alpha )}^{m}$ and ${(x\beta )}^{n}$, Jacobi polynomials and complexity
A previous article by Alin Bostan and colleagues last year described explicit expressions for the coefficients of the order$d$ polynomial subresultant of ${(x\alpha )}^{m}$ and ${(x\beta )}^{n}$ with respect to Bernstein's set of polynomials $\{{(x\alpha )}^{j}{(x\beta )}^{dj},\phantom{\rule{0.166667em}{0ex}}0\le j\le d\}$, for $0\le d<min\{m,n\}$. In 17, Alin Bostan, together with T. Krick, M. Valdettaro (U. Buenos Aires, Argentina) and A. Szanto (U. North Carolina, Raleigh, USA) further developed the study of these structured polynomials and showed that the coefficients of the subresultants of ${(x\alpha )}^{m}$ and ${(x\beta )}^{n}$ with respect to the monomial basis can be computed in linear arithmetic complexity, which is faster than for arbitrary polynomials. The result is obtained as a consequence of the amazing though seemingly unnoticed fact that these subresultants are scalar multiples of Jacobi polynomials up to an affine change of variables.
5.1.4 Improved algorithms for left factorial residues
In 11, Alin Bostan together with Vladica Andrejić (University of Belgrade, Serbia) and Milos Tatarevic (CoinList, Alameda, CA) presented improved algorithms for computing the left factorial residues $!p=0!+1!+\cdots +(p1)!\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}p$. They used these algorithms for the calculation of the residues $!p\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}p$, for all primes $p$ up to ${2}^{40}$. Their results confirm that Kurepa’s left factorial conjecture is still an open problem, as they show that there are no odd primes $p<{2}^{40}$ such that $p$ divides $!p$. Additionally, they confirmed that there are no socialist primes $p$ with $5<p<{2}^{40}$.
5.1.5 Explicit degree bounds for right factors of linear differential operators
If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominators of very high degree. When the base field is $\u2102$, Alin Bostan together with Bruno Salvy (Inria and ENS Lyon) and Tanguy Rivoal (CNRS and U. Grenoble) gave in 18 a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as the largest modulus of the local exponents at all its singularities. As a consequence, if a differential operator $L$ has rational function coefficients over a number field, they obtain degree bounds for its monic right factors in terms of the degree, the order and the height of $L$, and of the degree of the number field.
5.2 Polynomial systems and nonlinear equations
5.2.1 Complexity analysis of structured polynomial system solving
With Peter Bürgisser (Technische Universität Berlin, Germany) and Felipe Cucker (City University of Hong Kong), Pierre Lairez has developed an algorithm to compute an approximate zero of a polynomial system given as a blackbox evaluation function. Based on this, they study the average complexity of solving polynomial systems with low evaluation complexity.
Previous average complexity analyses of numerical algorithms to solve polynomial systems assume a dense model of random polynomials, far from the applications. In the work 31, Pierre Lairez and his colleagues deal with a model of random polynomials (random algebraic branching programs, ABP) indexed by the evaluation complexity and coming from a universal model of computation in algebraic complexity theory. They show that their algorithm performs $\mathrm{poly}(n,\delta )$ operations on average to solve random ABP in $n$ variables, of degree $\delta $ and of size $\mathrm{poly}(n,\delta )$. This brings complexity analysis closer than ever to what happens in practice.
5.2.2 Separation of periods of quartic surfaces
Given two numbers $A$ and $B$, a high precision numerical computation (say 10000 digits) can convince anybody that $A=B$ if the numerical approximations agree. However, this is not a proof. A separation bound yields a theorem in the form “$A=B$ or $AB>\u03f5$”. With Emre Sertöz (formerly MPI Leipzig, Germany, now MPI Bonn), Pierre Lairez proved a separation bound for periods of quartic surfaces 35. This is a surprising result as not much is known about general families of periods (although much is known about specific periods like $\pi $). Moreover, it gives credit to numerical approaches in effective algebraic geometry. Indeed, the usual criticism is the idea that numerical methods are good heuristics but cannot possibly prove anything. This result is a downright refutation of this argument.
5.2.3 A Gröbnerbasis theory for divideandconquer recurrences
Divideandconquer recurrences relate values of a given sequence at indices of the form ${b}^{\ell}n+r$ for fixed $b\ge 2$ and various pairs $(\ell ,r)$. Using systems of such recurrences makes it possible to express the behaviour of a quantity according to congruence classes modulo ${b}^{\ell}$. The study of such systems is poorly developed so far. In particular, no consistency check is available and no theory of initial values has been developed yet. In an ongoing study in order to fill this gap, Frédéric Chyzak and Philippe Dumas have introduced a noncommutative multivariate polynomial setting to represent divideandconquer systems. This setting involves at the same time variables that behave like words in purely noncommutative algebras and variables governed by commutation rules like in skew polynomial extensions. In their work 25, they initiate the study of left ideals of such polynomials and they develop their Gröbnerbasis theory, including the usual division and Buchberger algorithms. Strikingly, the nature of commutations generally prevents the leading monomial of a polynomial product to be the product of the leading monomials. To overcome the difficulty, they consider a specific monomial ordering, together with a restriction to monic divisors in intermediate steps. They also develop a variant of the ${F}_{4}$ algorithm with distinguishing features.
5.2.4 The Sage package comb_walks for walks in the quarter plane
With Antonio JiménezPastor who was visiting the team during his doctoral preparation, Alin Bostan, Frédéric Chyzak, and Pierre Lairez worked on a new software library designed to work with generating functions that count walks in the quarter plane. With this library for the Sagemath system they offer a cohesive package that brings together all the required procedures for manipulating these generating functions, as well as a unified interface to deal with them. They also display on a public webpage results that this package computes. They presented their work as an extended abstract at the conference ISSAC 14.
5.2.5 Differential transcendence of Bell numbers and relatives: a Galoistheoretic approach
A power series is called differentially transcendent if it does not satisfy any algebraic differential equation. In 2003, Martin Klazar proved in an elementary but very clever way that the ordinary generating function of the famous combinatorial Bell numbers, counting partitions of sets, is differentially transcendent. In 27, Alin Bostan (of the team), together with Lucia Di Vizio (CNRS, Université de Versailles) and Kilian Raschel (CNRS, Université de Tours), showed that Klazar's result is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. In the paper, they present the main principles of this theory in order to prove a general result of differential transcendence, that they apply to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by Klazar. Most of their examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the PakYeliussizov conjecture, according to which a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a linear recurrence with polynomial coefficients.
5.2.6 On the exponential generating function of labelled trees
In 16, Alin Bostan and Antonio JiménezPastor (U. Linz, Austria), proved that the exponential generating function of labelled trees, $T\left(x\right)={\sum}_{n\ge 1}\frac{{n}^{n1}}{n!}{x}^{n}$, is not ${\mathrm{D}}^{\infty}$finite. In particular, this implies that, although $T\left(x\right)$ satisfies nonlinear differential equations, these equations cannot be “too simple”. In particular, $T\left(x\right)$ is not the quotient of two Dfinite functions (satisfying linear differential equations with polynomial coefficients), More generally, $T\left(x\right)$ does not satisfy any linear differential equation with Dfinite coefficients. The proof ultimately relies on a result in differential Galois theory. Several open questions are left, of which one concerning the nature of the ordinary generating function of labelled trees, ${\sum}_{n\ge 1}{n}^{n1}{x}^{n}$.
5.3 Applications to combinatorics and probability theory
5.3.1 Bijections between Łukasiewicz walks and generalized tandem walks
Frédéric Chyzak and Karen Yeats (University of Waterloo, Canada) have studied the enumeration by length of several walk models on the square lattice. In the work 32 published this year, they obtain bijections between walks in the upper halfplane returning to the $x$axis walks in the quarter plane. A recent unpublished work by Bostan, Chyzak, and Mahboubi had given a bijection for models using small north, west, and southeast steps. In this year's publication, Chyzak and Yeats adapt and generalize it to a bijection between halfplane walks using those three steps in two colours and a quarterplane model over the symmetrized step set consisting of north, northwest, west, south, southeast, and east. They then generalize their bijections to certain models with large steps: for given $p\ge 1$, a bijection is given between the halfplane and quarterplane models obtained by keeping the small southeast step and replacing the two steps north and west of length 1 by the $p+1$ steps of length $p$ in directions between north and west. This model is close to, but distinct from, the model of generalized tandem walks studied by BousquetMélou, Fusy, and Raschel.
5.3.2 Weaklyunambiguous Parikh automata and their link to holonomic series
In 22 Alin Bostan together with Arnaud Carayol, Florent Koechlin and Cyril Nicaud (Univ. MarnelaVallée, France) investigated the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. They first proved a strong version of a conjecture by Castiglione and Massazza: weaklyunambiguous Parikh automata are equivalent to unambiguous twoway reversal bounded counter machines, and their multivariate generating series are holonomic. They then show that the converse is not true: they construct a language whose generating series is algebraic (thus holonomic), but which is inherently weaklyambiguous as a Parikh automata language. Finally, they prove an effective decidability result for the inclusion problem for weaklyunambiguous Parikh automata, and provide an upperbound on its complexity.
5.3.3 Counting walks with large steps in an orthant
In the past fifteen years, the enumeration of lattice walks with steps taken in a prescribed set and confined to a given cone, especially the first quadrant of the plane, has been intensely studied. As a result, the generating functions of quadrant walks are now wellunderstood, provided the allowed steps are small. In particular, having small steps is crucial for the definition of a certain group of birational transformations of the plane. It has been proved that this group is finite if and only if the corresponding generating function is Dfinite. This group is also the key to the uniform solution of 19 of the 23 small step models possessing a finite group. In contrast, almost nothing was known for walks with arbitrary steps. In 12, Alin Bostan together with Mireille BousquetMélou (CNRS, Bordeaux) and Stephen Melczer (U. Pennsylvania, Philadelphia, USA), extended the definition of the group, or rather of the associated orbit, to this general case, and generalized the above uniform solution of small step models. When this approach works, it invariably yields a Dfinite generating function. They applied it to many quadrant problems, including some infinite families. After developing the general theory, the authors of 12 considered the $13\phantom{\rule{4pt}{0ex}}110$ twodimensional models with steps in ${\{2,1,0,1\}}^{2}$ having at least one $2$ coordinate. They proved that only 240 of them have a finite orbit, and solve 231 of them with their method. The 9 remaining models are the counterparts of the 4 models of the small step case that resist the uniform solution method (and which are known to have an algebraic generating function). They conjecture Dfiniteness for their generating functions (but only two of them are likely to be algebraic!), and proved nonDfiniteness for the $12\phantom{\rule{4pt}{0ex}}870$ models with an infinite orbit, except for 16 of them.
5.3.4 The generating function of Kreweras walks with interacting boundaries is not algebraic
Beaton, Owczarek and Xu (2019) studied generating functions of Kreweras walks and of reverse Kreweras walks in the quarter plane, with interacting boundaries. They proved that for the reverse Kreweras step set, the generating function is always algebraic, and for the Kreweras step set, the generating function is always Dfinite. However, apart from the particular case where the interactions are symmetric in $x$ and $y$, they left open the question of whether the latter one is algebraic. Using computer algebra tools, Alin Bostan, together with Manuel Kauers and Thibaut Verron (University, Linz, Austria) confirmed 28 the previous intuition that the generating function of Kreweras walks is not algebraic, apart from the particular case already identified.
5.3.5 Martin boundary of killed random walks on isoradial graphs
Alin Bostan contributed to an article by C. Boutillier (Sorbonne Université) and K. Raschel (CNRS, Université de Tours) 19, devoted to the study of random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially nonhomogeneous. However, Boutillier and Raschel have been able to obtain analogues of a celebrated result by Ney and Spitzer (1966) on the socalled Martin kernel (ratio of Green functions started at different points). Alin Bostan provided in the Appendix two different proofs of the fact that some algebraic power series arising in this context have nonnegative coefficients.
5.3.6 Stieltjes moment sequences for patternavoiding permutations
A small subset of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0,\infty )$. Such sequences are known as Stieltjes moment sequences. They have a number of useful properties, such as logconvexity, which in turn enables one to rigorously bound their growth constant from below.
In 15, Alin Bostan together with Andrew Elvey Price (Université de Bordeaux), Anthony Guttmann (University of Melbourne), and JeanMarie Maillard (Sorbonne Université), studied some classical sequences in enumerative combinatorics, denoted $Av\left(\mathcal{P}\right)$, and counting permutations of $\{1,2,...,n\}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12...k)$, they showed that the corresponding sequences, $Av(123...k)$, are Stieltjes moment sequences, and explicitly determined the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool.
They showed that the densities for $\phantom{\rule{0.166667em}{0ex}}Av\left(1234\right)$ and $\phantom{\rule{0.166667em}{0ex}}Av\left(12345\right)$, correspond to an orderone linear differential operator acting on a classical modular form given as a pullback of a Gaussian ${\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}$ hypergeometric function, respectively to an ordertwo linear differential operator acting on the square of a classical modular form given as a pullback of a ${\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}$ hypergeometric function. Moreover, these density functions are closely, but nontrivially, related to the density attached to the distance traveled by a walk in the plane with $k1$ unit steps in random directions.
As a bonus, they studied the challenging case of the $Av\left(1324\right)$ sequence and gave compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, they proved new lower bounds on the growth constant of this sequence, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give a good estimate of the (unknown) growth constant.
5.3.7 A note on gamma triangles and local gamma vectors
Alin Bostan contributed to F. Chapoton's article 20 by writing an appendix, which allowed the author to complete its article. The theme of 20 is the study of simplicial complexes in algebraic combinatorics. A basic invariant is the $f$vector that counts faces according to their dimensions. A less understood invariant is the $\gamma $vector, introduced by Gal in 2005. Also in 2005, Chapoton, motivated by the study of the combinatorics of simplicial complexes attached to cluster algebras, considered a refined version of the $f$vector. The main aim of 20 is to introduce the analogue in this context of the $\gamma $vector, and a further refinement called the $\Gamma $triangle. The author computed explicitly the $\Gamma $triangle for all the cluster simplicial complexes of irreducible Coxeter groups. Alin Bostan contributed to the proof of an unexpected relation between the $\Gamma $triangles of cluster fans of type $\mathbb{B}$ and $\mathbb{D}$.
5.3.8 On an integral identity
In 13, Alin Bostan together with Fernando Chamizo (Universidad Autónoma de Madrid and ICMAT, Spain) and Mikael Persson Sundqvist (Lund University, Sweden) gave three elementary proofs of a nice equality of definite integrals, recently proven by Ekhad, Zeilberger and Zudilin. The equality arises in the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. They furthermore provide a generalization together with an equally elementary proof and discuss some consequences.
5.3.9 Diagonal representation of algebraic power series: a glimpse behind the scenes
There are many viewpoints on algebraic power series, ranging from the abstract ringtheoretic notion of Henselization to the very explicit perspective as diagonals of certain rational functions. Denef and Lipshitz proved in 1987 that any algebraic power series in $n$ variables can be written as a diagonal of a rational power series in one variable more. Their proof uses a lot of involved theory and machinery which remains hidden to the reader in the original article. In the work 26, which is based on his master's thesis, Sergey Yurkevich explained these tools by motivating while defining them and reproving most of their interesting parts. Moreover, he provided a new significant improvement on the ArtinMazur lemma, proving the existence of a 2dimensional code of algebraic power series.
5.3.10 On a class of hypergeometric diagonals
In 30, Alin Bostan together with his PhD student Sergey Yurkevich proved that the diagonal of any finite product of algebraic functions of the form
is a generalized hypergeometric function, and they provided explicit description of its parameters. The particular case ${(1xy)}^{R}/(1xyz)$ corresponds to the main identity of Abdelaziz, Koutschan and Maillard in 36. The result in 30 is useful in both directions: on the one hand it shows that Christol's conjecture holds true for a large class of hypergeometric functions, on the other hand it allows for a very explicit and general viewpoint on the diagonals of algebraic functions of the type above. Finally, in contrast to 36, the new proof is completely elementary and does not require any algorithmic help.
5.3.11 Genus and classification of random walks in the quarter plane
In collaboration with R. Iasnogorodski (SPCPA, SaintPetersburg), Guy Fayolle analyzes the kernel$K(x,y,t)$ of the basic functional equation associated with the trivariate counting generating function (CGF) of walks in the quarter plane. In this short paper 21, taking $t\in ]0,1[$, we provide the conditions on the step set $\left\{{p}_{i,j}\right\}$ to decide whether the walks are singular or regular, as defined in 57. These conditions are independent of $t\in ]0,1[$ and given in terms of step set configurations. They also find the configurations for the kernel to be of genus 0, knowing that the genus is always $\le 1$. All these conditions are very similar to the case $t=1$ considered in 57. Their results extend the work
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5.3.12 Reflected Brownian motion in a nonconvex cone
In an ongoing work in collaboration with S. Franceschi (LMO, ParisSaclay University) and K. Raschel (CNRS, Tours University), Guy Fayolle states a system of functional equations satisfied by the Laplace transform of the stationary distribution of a reflected Brownian motion (SRBM) in a twodimensional nonconvex cone. While the case of convex cones is now reasonably well studied, the framework of nonconvex cones turns out to be more challenging, as shown by similar research carried out in a discrete setting. They show in particular that the problem can be reduced to a boundary value problem of Rieman–Hilbert–Carleman type on an hyperbola, for a twodimensional vector of meromorphic functions. This seems to be a quite original result.
5.3.13 Random walks in orthants and lattice path combinatorics
In the second edition of the book 57, original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps (size 1), the general solution being obtained via reduction to boundary value problems. Among other things, an important quantity, the socalled group of the walk, allows to deduce theoretical features about the nature of the solutions. In particular, when the order of the group is finite and the underlying algebraic curve is of genus 0 or 1, necessary and sufficient conditions have been given for the solution to be rational, algebraic or $D$finite (i.e. solution of a linear differential equation). In this framework, a number of difficult open problems related to lattice path combinatorics are currently being explored boundary Alin Bostan, Frédéric Chyzak, and Guy Fayolle, both from theoretical and computer algebra points of view: concrete computation of the criteria, utilization of differential Galois theory, genus greater than 1 (i.e., when some jumps are of size $\ge 2$), etc. This relates simple productform stochastic networks (socalled Jackson networks) and explicit solutions of functional equations for counting lattice walks. Some partial extensions of 33 are under development.
5.4 Formal methods
5.4.1 Recursive operator definitions
TLA+ originally allowed recursive function definitions, but not recursive operator definitions, because it was not known how how to define their semantics. They were added to the language in 2006 after a semantics was discovered for them. This year, Georges Gonthier, together with Leslie Lamport (Microsoft Research, USA), described that semantics in 34.
6 Partnerships and cooperations
6.1 International research visitors
6.1.1 Visits to international teams
Research stays abroad
 Alin Bostan visited (March 2020) Herwig Hauser's team at the University of Vienna. The initial plan was that Alin Bostan would be invited professor from March to June 2020, and that he would teach a Master course titled “Experimental Mathematics and Computer Algebra for Combinatorics and Number Theory”. Unfortunately, his research stay and his lectures had to be aborted after few weeks due to the COVID19 pandemic, and postponed.
6.2 National initiatives
6.2.1 ANR
 De rerum natura. This project, set up by the team, was accepted this year and will be funded until 2023. It gathers over 20 experts from four fields: computer algebra; the Galois theories of linear functional equations; number theory; combinatorics and probability. Our goal is to obtain classification algorithms for number theory and combinatorics, particularly so for deciding irrationality and transcendence. (Permanent members with pm listed: Bostan, Chyzak, Lairez.)
 $\partial $ifference. This project, led by Olivier Bournez (Lix), started in November 2020. Its objective is to consider a novel approach in between the two worlds: discreteoriented computations on the one side and differential equations on the other side. We aims at providing new insights on classical complexity theory, computability and logic through this prism and at introducing new perspectives in algorithmic methods for differential equations solving and computer science applications. (Permanent members with pm listed: Bostan, Chyzak.)
 Tremplin ERC. Pierre Lairez has been awarded a “tremplin” project by ANR. This will help him prepare an ERC project submission “10000 Digits, Foundations of transcendental methods in numerical algebraic geometry”.
6.3 Regional initiatives
 Alin Bostan submitted in November 2020 a PCRIANR proposal ELEFANT – “Efficient aLgorithms For guessing, summAtioN and posiTivity”. This is a bilateral ANR/FWF project between 2 computer algebra teams in France and 2 computer algebra teams in Austria. The Austrian coleader is Manuel Kauers from Univ. Linz. The goal is to work together on four axes: structured and multivariate guessing, positivity and Dfiniteness, creative telescoping and applications in combinatorics, number theory and theoretical physics. The requested funding is of 770,000 euros in total.
 Alin Bostan is coleader of the bilateral project “Integer Sequences arising in Number Theory, Combinatorics and Physics” between France and Austria. The Austrian coleader is Herwig Hauser (U. Vienna, Austria).
7 Dissemination
7.1 Promoting scientific activities
7.1.1 Scientific events: organisation
General chair, scientific chair
 Alin Bostan is part of the Scientific advisory board of the conference series Effective Methods in Algebraic Geometry (MEGA).
 Since 2020, for a period of 5 years, Alin Bostan is member of the steering committee of the Journées Nationales de Calcul Formel (JNCF), the annual meeting of the French computer algebra community.
 Alin Bostan is part of the scientific committee of the GDR EFI (“Functional Equations and Interactions”) dependent on the mathematical institute (INSMI) of the CNRS. The goal of this GDR is to bring together various research communities in France working on functional equations in fields of computer science and mathematics.
 Frédéric Chyzak is General Chair of the international conference ISSAC'21 (International Symposium on Symbolic and Algebraic Computation), whose organizing started in 2020.
Member of the organizing committees
 Alin Bostan coorganizes, with Lucia Di Vizio, the Séminaire Différentiel between U. Versailles and Inria Saclay, with a biannual frequency.
 Alin Bostan coorganizes, with Lucia Di Vizio and Kilian Raschel the working group Transcendance et Combinatoire, at Institut Henri Poincaré (Paris), with a bimonthly frequency.
 Alin Bostan, together with Mohab Safey El Din, Bruno Salvy and Gilles Villard, wrote a proposal for a thematic program “Recent Trends in Computer Algebra (RTCA)”, to be held in 2023 in Paris and Lyon. The proposal has been accepted, the main funders being IHP (120,000 euros) and Labex Milyon (60,000 euros).
7.1.2 Scientific events: selection
Member of the conference program committees
 Pierre Lairez has served in the program committee of ISSAC'20 (International Symposium on Symbolic and Algebraic Computation).
Reviewer
 Frédéric Chyzak has been a reviewer for ISSAC'20 (International Symposium on Symbolic and Algebraic Computation), CASC'20 (Computer Algebra in Scientific Computing), and for a postproceeding of Transient Transcendence in Transylvania.
 Alin Bostan has been a reviewer for ISSAC'20 (International Symposium on Symbolic and Algebraic Computation) and AofA'20 (Analysis of Algorithms).
7.1.3 Journal
Member of the editorial boards
 Alin Bostan is on the editorial board of the Journal of Symbolic Computation.
 Alin Bostan is on the editorial board of the Annals of Combinatorics
 Frédéric Chyzak is on the editorial board of the Journal of Systems Science and Complexity.
 Guy Fayolle is associate editor of the journal Markov Processes and Related Fields (MPRF).
Reviewer  reviewing activities
 In 2020, Frédéric Chyzak has been a reviewer for Journal of Symbolic Computation, Transactions of Mathematical Software, Journal de l'École polytechnique, Mathématiques, and for International Journal of Number Theory.
 Guy Fayolle has been a reviewer for Advances in Applied Probability, Markov Processes and Related Fields, Probability Theory and Related Fields, Queueing Systems: Theory and Applications, European Journal of Combinatorics, Journal of Statistical Physics, Physica A, Springer Science.
 In 2020, Alin Bostan has been a reviewer for Mathematics of Computation, Experimental Mathematics, Journal of Combinatorial Theory, Series A, Séminaire Lotharingien de Combinatoire, Journal of Algebraic Combinatorics, Bulletin de la Société Mathématique de France, Glasgow Mathematical Journal, Advances in Applied Mathematics, Journal of Applied Analysis.
7.1.4 Invited talks
 Frédéric Chyzak has been invited to present his work on symbolic integration 5 at the 10th international workshop on Differential Algebra and Related Topics (DART X) (New York, USA) and in the Algorithmic Combinatorics Seminar (Research Institute for Symbolic Computation, Linz).
7.1.5 Scientific expertise
 Guy Fayolle is scientific advisor and associate researcher at the Robotics Laboratory of Mines ParisTech.
7.1.6 Research administration
 Guy Fayolle is a member for Computer System Modeling of the International Federation for Information Processing (IFIP WG 7.3).
7.2 Teaching  Supervision  Juries
7.2.1 Teaching

Bachelor:
 Alexandre Goyer, Mathématiques Générales (LSMA100), 54h, L1, Université de Versailles SaintQuentinenYvelines, France.

Master:
 Alin Bostan, Algorithmes efficaces en calcul formel, 36h, M2, MPRI, France.
 Alin Bostan, Modern Algorithms for Symbolic Summation and Integration, 18h, M2, Master d'Informatique Fondamentale de l'ENS de Lyon, France.
 Frédéric Chyzak, Algorithmes efficaces en calcul formel, 36h, M2, MPRI, France. (Also responsible for the course.)
 Pierre Lairez, Algorithmique avancée (INF550), TD, 18h, M2, École polytechnique, France.
 Pierre Lairez, Les bases de la programmation et de l'algorithmique (INF411), TD, 40h, M1, École polytechnique, France.
7.2.2 Supervision

Bachelor internships:
 Frédéric Chyzak cosupervised together with Jérémy Berthomieu (Sorbonne Université) the Bachelor internship of Lucas Morisset on the topic “Fast manipulation of polynomial systems: the ${F}_{5}$ algorithm”.

Master interships:
 Alin Bostan cosupervised together with Xavier Caruso (CNRS, IMB Bordeaux) the Master thesis of Raphaël Pagès on the topic “Computing characteristic polynomials of pcurvatures in average polynomial time”. The result was submitted for publication to ISSAC 2021.
 Frédéric Chyzak cosupervised together with Marc Mezzarobba (CNRS) the Master thesis of Alexandre Goyer on the topic “Symbolicnumeric algorithm for the factorization of differential operators”.

PhD theses:
 Alin Bostan cosupervises together with Xavier Caruso (CNRS, IMB Bordeaux) the PhD thesis of Raphaël Pagès on the topic “Algorithms for factoring linear differential operators in positive characteristic”.
 Alin Bostan cosupervises together with Herwig Hauser (U. Vienna, Austria) the PhD thesis of Sergey Yurkevich on the topic “Integer Sequences arising in Number Theory, Combinatorics and Physics”.
 Frédéric Chyzak cosupervises together with Marc Mezzarobba (CNRS, Lix, Palaiseau) the PhD thesis of Alexandre Goyer on the topic “Symbolicnumeric algorithms in differential algebra”.
7.2.3 Juries
 Alin Bostan has served as an examiner in the PhD jury of Youssef Abdelaziz, Diagonales de fractions rationnelles en physique, Sorbonne Univ., September 18, 2020.
 Alin Bostan has served as a member of the monitoring PhD committee of Youssef Abdelaziz, Sorbonne Univ.
 Alin Bostan has served as a member of the monitoring PhD committee of Manon Bertin, Univ. Rouen.
 Alin Bostan has served as a member of the monitoring PhD committee of Isabella Panaccione, Ecole polytechnique.
 Frédéric Chyzak has served as a reviewer in the PhD jury of Antonio Jiménez Pastor, A computable extension for holonomic functions: DDfinite functions, Johannes Kepler University Linz, December 9, 2020.
 Frédéric Chyzak has served as a reviewer in the midPhD examination of Mathilde Chenu, Study of isogenybased primitives for postquantum cryptography.
8 Scientific production
8.1 Major publications
 1 inproceedingsQuasioptimal multiplication of linear differential operatorsFOCS 2012  IEEE 53rd Annual Symposium on Foundations of Computer ScienceNew Brunswick, United StatesIEEEOctober 2012, 524530
 2 article Counting walks with large steps in an orthant Journal of the European Mathematical Society 2020
 3 inproceedingsFast Coefficient Computation for Algebraic Power Series in Positive CharacteristicANTSXIII  Thirteenth Algorithmic Number Theory Symposium2Proceedings of the Thirteenth Algorithmic Number Theory Symposium (ANTSXIII)1Madison, United StatesMathematical Sciences PublishersJuly 2018, 119135
 4 book Algorithmes Efficaces en Calcul Formel Voir la page du livre à l'adresse https://hal.archivesouvertes.fr/AECF/ published by the Authors 2017
 5 inproceedingsGeneralized Hermite Reduction, Creative Telescoping and Definite Integration of DFinite FunctionsISSAC 2018  International Symposium on Symbolic and Algebraic ComputationNew York, United StatesJuly 2018, 18
 6 articleHypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter PlaneEuropean Journal of Combinatorics612017, 242275
 7 articleComputing the Homology of Basic Semialgebraic Sets in Weak Exponential TimeJournal of the ACM (JACM)661December 2018, 130
 8 articleComputing solutions of linear Mahler equationsMathematics of Computation87July 2018, 29773021

9
inproceedings
A ComputerAlgebraBased Formal Proof of the Irrationality of
$$ (3)' ITP  5th International Conference on Interactive Theorem Proving Vienna, Austria 2014  10 articleComputing periods of rational integralsMathematics of Computation85November 2016, 17191752
8.2 Publications of the year
International journals
 11 article Improved algorithms for left factorial residues Information Processing Letters 2021
 12 article Counting walks with large steps in an orthant Journal of the European Mathematical Society 2020
 13 article On an Integral Identity American Mathematical Monthly 2020
 14 articleThe Sage Package comb_walks for Walks in the Quarter PlaneACM Communications in Computer Algebra542122020, 9
 15 articleStieltjes moment sequences for patternavoiding permutationsThe Electronic Journal of Combinatorics2020, 59
 16 article On the exponential generating function of labelled trees Comptes rendus de l'Académie des sciences. Série I, Mathématique 2020

17
articleSubresultants of
${(x)}^{m}$ and${(x)}^{n}$ , Jacobi polynomials and complexity'Journal of Symbolic Computation1012020, 330351  18 articleExplicit degree bounds for right factors of linear differential operatorsBulletin of the London Mathematical Society531February 2021, 5362
 19 article Martin boundary of killed random walks on isoradial graphs Potential Analysis 2021
 20 articleA note on gamma triangles and local gamma vectorsAnnales de la Faculté des Sciences de Toulouse. Mathématiques.4December 2020, 907925
 21 article Conditions for some non stationary random walks in the quarter plane to be singular or of genus 0 Markov Processes And Related Fields March 2021
International peerreviewed conferences
 22 inproceedingsWeaklyunambiguous Parikh automata and their link to holonomic seriesICALP 2020  47th International Colloquium on Automata, Languages and ProgrammingSaarbrücken, Germanyhttps://icalp2020.saarlandinformaticscampus.deJuly 2020, 16

23
inproceedingsComputing the
$N$ th Term of a$q$ Holonomic Sequence'ISSAC'20: Proceedings of the 45th International Symposium on Symbolic and Algebraic ComputationISSAC 2020  45th International Symposium on Symbolic and Algebraic ComputationKalamata, GreeceJuly 2020, 8 
24
inproceedings
A Simple and Fast Algorithm for Computing the
$N$ th Term of a Linearly Recurrent Sequence' SOSA'21 (SIAM Symposium on Simplicity in Algorithms) Alexandria, United States https://www.siam.org/conferences/cm/conference/sosa21 January 2021  25 inproceedings A GröbnerBasis Theory for DivideandConquer Recurrences ISSAC'20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation ISSAC  2020  45th International Symposium on Symbolic and Algebraic Computation Kalamata, Greece July 2020
 26 inproceedings Diagonal Representation of Algebraic Power Series: A Glimpse Behind the Scenes Transient Transcendence in Transylvania This work is based on the author's master's thesis (U. of Vienna, 2020), supervised by H. Hauser. Brașov, Romania May 2019
Reports & preprints
 27 misc Differential transcendence of Bell numbers and relatives: a Galois theoretic approach December 2020
 28 misc The generating function of Kreweras walks with interacting boundaries is not algebraic 2020

29
misc
Fast Computation of the
$N$ th Term of a$q$ Holonomic Sequence and Applications' 2020  30 misc On a Class of Hypergeometric Diagonals 2020
 31 misc Rigid continuation paths II. Structured polynomial systems October 2020
 32 misc Bijections between Łukasiewicz walks and generalized tandem walks February 2020
 33 misc A note on the connection between productform Jackson networks and counting lattice walks in the quarter plane January 2020
 34 reportRecursive Operator DefinitionsInria Saclay Ile de FranceMay 2020, 17
 35 misc Separation of periods of quartic surfaces December 2020
8.3 Cited publications
 36 articleOn Christol's conjectureJ. Phys. A53202020, 205201, 16 pagesURL: https://doi.org/10.1088/17518121/ab82dc
 37 bookM. AbramowitzI. StegunHandbook of mathematical functions with formulas, graphs, and mathematical tablesReprint of the 1972 editionNew YorkDover1992, xiv+1046
 38 inproceedings Extending Coq with Imperative Features and its Application to SAT Verication Interactive Theorem Proving, international Conference, ITP 2010, Edinburgh, Scotland, July 1114, 2010, Proceedings Lecture Notes in Computer Science Springer 2010
 39 articleA uniform approach for the fast computation of matrixtype Padé approximantsSIAM J. Matrix Anal. Appl.1531994, 804823
 40 inproceedingsThe Dynamic Dictionary of Mathematical Functions (DDMF)The Third International Congress on Mathematical Software (ICMS 2010)6327Lecture Notes in Computer Science2010, 3541
 41 inproceedings Full reduction at full throttle First International Conference on Certified Programs and Proofs, Taiwan, December 79 Lecture Notes in Computer Science Springer 2011
 42 incollectionImproving Real Analysis in Coq: A UserFriendly Approach to Integrals and DerivativesCertified Programs and Proofs7679Lecture Notes in Computer ScienceSpringer Berlin Heidelberg2012, 289304URL: http://dx.doi.org/10.1007/9783642353086_22
 43 inproceedingsFlocq: A Unified Library for Proving Floatingpoint Algorithms in CoqProceedings of the 20th IEEE Symposium on Computer ArithmeticTübingen, GermanyJuly 2011, 243252
 44 inproceedingsAlgorithmes rapides pour les polynômes, séries formelles et matricesActes des Journées Nationales de Calcul FormelLes cours du CIRM, tome 1, numéro 2Luminy, France2010, 75262URL: http://ccirm.cedram.org:80/ccirmbin/fitem?id=CCIRM_2010__1_2_75_0
 45 articleGlobally nilpotent differential operators and the square Ising modelJ. Phys. A: Math. Theor.42122009, 50URL: http://dx.doi.org/10.1088/17518113/42/12/125206
 46 inproceedingsComplexity of creative telescoping for bivariate rational functionsISSAC'10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic ComputationNew York, NY, USAACM2010, 203210URL: http://doi.acm.org/10.1145/1837934.1837975
 47 inproceedingsDifferential equations for algebraic functionsISSAC'07: Proceedings of the 2007 international symposium on Symbolic and algebraic computationACM Press2007, 2532URL: http://dx.doi.org/10.1145/1277548.1277553
 48 articleExplicit formula for the generating series of diagonal 3D rook pathsSém. Loth. Comb.B66a2011, 27URL: http://www.emis.de/journals/SLC/wpapers/s66bochhope.html
 49 articleThe complete generating function for Gessel walks is algebraicProceedings of the American Mathematical Society1389With an appendix by Mark van Hoeij2010, 30633078
 50 articleAn extension of Zeilberger's fast algorithm to general holonomic functionsDiscrete Math.21713Formal power series and algebraic combinatorics (Vienna, 1997)2000, 115134
 51 inproceedingsA NonHolonomic Systems Approach to Special Function IdentitiesISSAC'09: Proceedings of the TwentySecond International Symposium on Symbolic and Algebraic Computation2009, 111118URL: http://dx.doi.org/10.1145/1576702.1576720
 52 articleNoncommutative elimination in Ore algebras proves multivariate identitiesJ. Symbolic Comput.2621998, 187227
 53 misc Computer Algebra Errors URL: http://mathoverflow.net/questions/11517/computeralgebraerrors
 54 articleThe Calculus of ConstructionsInf. Comput.762/31988, 95120URL: http://dx.doi.org/10.1016/08905401(88)900053
 55 inproceedings Inductively defined types Proceedings of Colog'88 417 Lecture Notes in Computer Science SpringerVerlag 1990
 56 articleDealing with algebraic expressions over a field in Coq using MapleJ. Symbolic Comput.395Special issue on the integration of automated reasoning and computer algebra systems2005, 569592URL: http://dx.doi.org/10.1016/j.jsc.2004.12.004
 57 book Random walks in the quarter plane Springer International Publishing 2017
 58 inproceedingsPackaging Mathematical StructuresTheorem Proving in HigherOrder Logics5674Lecture Notes in Computer ScienceSpringer2009, 327342
 59 articleFormal proofsthe fourcolour theoremNotices of the AMS55112008, 13821393
 60 articleAn introduction to small scale reflection in CoqJournal of Formalized Reasoning322010, 95152
 61 techreport A Small Scale Reflection Extension for the Coq system RR6455 INRIA 2008
 62 inproceedingsA language of patterns for subterm selectionITP7406LNCS2012, 361376
 63 inproceedingsProving Equalities in a Commutative Ring Done Right in CoqTheorem Proving in Higher Order Logics, 18th International Conference, TPHOLs 2005, Oxford, UK, August 2225, 2005, Proceedings3603Lecture Notes in Computer ScienceSpringer2005, 98113
 64 articleFormal proofNotices of the AMS55112008, 13701380
 65 inproceedingsA HOL Theory of Euclidean spaceTheorem Proving in Higher Order Logics, 18th International Conference, TPHOLs 20053603Lecture Notes in Computer ScienceOxford, UKSpringerVerlag2005, 114129
 66 inproceedingsA MachineChecked Theory of Floating Point ArithmeticTheorem Proving in Higher Order Logics: 12th International Conference, TPHOLs'991690Lecture Notes in Computer ScienceNice, FranceSpringerVerlag1999, 113130
 67 articleFormalizing an analytic proof of the prime number theoremJournal of Automated Reasoning43Dedicated to Mike Gordon on the occasion of his 60th birthday2009, 243261
 68 book Theorem proving with the real numbers CPHC/BCS distinguished dissertations Springer 1998
 69 articleA Skeptic's Approach to Combining HOL and MapleJ. Autom. Reason.213December 1998, 279294URL: http://dx.doi.org/10.1023/A:1006023127567
 70 miscAnother Mathematica bug2009, URL: http://fredrikj.blogspot.fr/2009/07/anothermathematicabug.html
 71 articleA fast approach to creative telescopingMath. Comput. Sci.4232010, 259266URL: http://dx.doi.org/10.1007/s1178601000550
 72 articleImplementing the cylindrical algebraic decomposition within the Coq systemMathematical Structures in Computer Science1712007, 99127
 73 article Mizar: the first 30 years Mechanized Mathematics and Its Applications 4 2005
 74 phdthesis Problèmes critiques et preuves formelles Université Paris 13 2012
 75 incollectionNumGfun: a package for numerical and analytic computation and Dfinite functionsISSAC 2010Proceedings of the 2010 International Symposium on Symbolic and Algebraic ComputationNew YorkACM2010, 139146URL: http://dx.doi.org/10.1145/1837934.1837965
 76 book F. Olver D. Lozier R. Boisvert C. Clark NIST Handbook of mathematical functions Cambridge University Press 2010

77
articleA Mathematica version of Zeilberger's algorithm for proving binomial coefficient identitiesJ. Symbolic Comput.2056Symbolic computation in combinatorics
${}_{1}$ (Ithaca, NY, 1993)1995, 673698URL: http://dx.doi.org/10.1006/jsco.1995.1071  78 misc Maple
 79 inproceedingsOn the Integrity of a Repository of Formalized MathematicsProceedings of the Second International Conference on Mathematical Knowledge ManagementMKM '03London, UKSpringerVerlag2003, 162174URL: http://dl.acm.org/citation.cfm?id=648071.748518
 80 articleGfun: a Maple package for the manipulation of generating and holonomic functions in one variableACM Trans. Math. Software2021994, 163177
 81 book The Encyclopedia of Integer Sequences Academic Press, San Diego 1995
 82 miscA Formalization of the Odd Order Theorem using the Coq proof assistant2012, URL: http://www.msrinria.fr/projects/mathematicalcomponents/
 83 misc The Coq Proof Assistant: Reference Manual URL: http://coq.inria.fr/doc/
 84 articleA MachineChecked Implementation of Buchberger's AlgorithmJ. Autom. Reasoning2622001, 107137URL: http://dx.doi.org/10.1023/A:1026518331905
 85 mastersthesisComputer generated proofs of binomial multisum identitiesMA ThesisRISC, J. Kepler UniversityMay 1997, 99
 86 book Mathematica: A system for doing mathematics by computer (2nd ed.) AddisonWesley 1992
 87 articleA holonomic systems approach to special functions identitiesJ. Comput. Appl. Math.3231990, 321368
 88 miscOpinion 94: The Human Obsession With “Formal Proofs” is a Waste of the Computer's Time, and, Even More Regretfully, of Humans' Time2009, URL: http://www.math.rutgers.edu/~zeilberg/Opinion94.html
 89 articleThe method of creative telescopingJ. Symbolic Comput.1131991, 195204
 90 bookModern computer algebraNew YorkCambridge University Press2003, xiv+785