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##### POLSYS - 2013

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## Section: New Results

### Fundamental Algorithms and Structured Systems

#### Structured polynomial systems: the quasi-homogeneous case

Let $𝕂$ be a field and $\left({f}_{1},...,{f}_{n}\right)\subset 𝕂\left[{X}_{1},...,{X}_{n}\right]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $\left({d}_{1},...,{d}_{n}\right)$ w.r.t a system of weights $\left({w}_{1},\cdots ,{w}_{n}\right)$. Such systems are likely to arise from a lot of applications, including physics or cryptography. In [29] , we design strategies for computing Gröbner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi-homogeneous system, the complexity of the full strategy is polynomial in the weighted Bézout bound ${\prod }_{i=1}^{n}{d}_{i}/{\prod }_{i=1}^{n}{w}_{i}$. We provide some experimental results based on generic systems as well as systems arising from a cryptography problem. They show that taking advantage of the quasi-homogeneous structure of the systems allow us to solve systems that were out of reach otherwise.

#### Structured polynomial systems: the determinantal case

In [13] , We study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a determinantal ideal: the ideal generated by all minors of size $r+1$ of the matrix. We give new complexity bounds for solving this problem using Gröbner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0-dimensional and radical system of bi-degree $\left(D,1\right)$. We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.

#### On the Complexity of the Generalized MinRank Problem

In [13] we study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a determinantal ideal: the ideal generated by all minors of size $r+1$ of the matrix. We give new complexity bounds for solving this problem using Gröbner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0-dimensional and radical system of bi-degree $\left(D,1\right)$. We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.

#### On the Complexity of Computing Gröbner Bases for Quasi-homogeneous Systems

Let $𝕂$ be a field and $\left({f}_{1},...,{f}_{n}\right)\subset 𝕂\left[{X}_{1},...,{X}_{n}\right]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $\left({d}_{1},...,{d}_{n}\right)$ w.r.t a system of weights $\left({w}_{1},\cdots ,{w}_{n}\right)$. Such systems are likely to arise from a lot of applications, including physics or cryptography.

In [29] , we design strategies for computing Gröbner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi-homogeneous system, the complexity of the full strategy is polynomial in the weighted Bézout bound ${\prod }_{i=1}^{n}{d}_{i}/{\prod }_{i=1}^{n}{w}_{i}$.

We provide some experimental results based on generic systems as well as systems arising from a cryptography problem. They show that taking advantage of the quasi-homogeneous structure of the systems allow us to solve systems that were out of reach otherwise.

#### Gröbner bases of ideals invariant under a commutative group : the non-modular case

In [30] , we propose efficient algorithms to compute the Gröbner basis of an ideal $I\subset k\left[{x}_{1},\cdots ,{x}_{n}\right]$ globally invariant under the action of a commutative matrix group $G$, in the non-modular case (where $char\left(k\right)$ doesn't divide $|G|$). The idea is to simultaneously diagonalize the matrices in $G$, and apply a linear change of variables on $I$ corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on $I$ are diagonal. This action induces a grading on the ring $R=k\left[{x}_{1},\cdots ,{x}_{n}\right]$, compatible with the degree, indexed by a group related to $G$, that we call $G$-degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into $|G|$ submatrices of roughly the same size. In the same way, we are able to split the canonical basis of $R/I$ (the staircase) if $I$ is a zero-dimensional ideal. Therefore, we derive abelian versions of the classical algorithms ${F}_{4}$, ${F}_{5}$ or FGLM. Moreover, this new variant of ${F}_{4}/{F}_{5}$ allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of ${F}_{4}$. Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time.

#### Signature Rewriting in Gröbner Basis Computation

In [27] we introduce the RB algorithm for Gröbner basis computation, a simpler yet equivalent algorithm to F5GEN. RB contains the original unmodified F5 algorithm as a special case, so it is possible to study and understand F5 by considering the simpler RB. We present simple yet complete proofs of this fact and of F5's termination and correctness. RB is parametrized by a rewrite order and it contains many published algorithms as special cases, including SB. We prove that SB is the best possible instantiation of RB in the following sense. Let X be any instantiation of RB (such as F5). Then the S-pairs reduced by SB are always a subset of the S-pairs reduced by X and the basis computed by SB is always a subset of the basis computed by X.

#### An analysis of inhomogeneous signature-based Gröbner basis computations

In [8] we give an insight into the behaviour of signature-based Gröbner basis algorithms, like F5, G2V or SB, for inhomogeneous input. On the one hand, it seems that the restriction to sig-safe reductions puts a penalty on the performance. The lost connection between polynomial degree and signature degree can disallow lots of reductions and can lead to an overhead in the computations. On the other hand, the way critical pairs are sorted and corresponding s-polynomials are handled in signature- based algorithms is a very efficient one, strongly connected to sorting w.r.t. the well-known sugar degree of polynomials.

#### Improving incremental signature-based Gröbner basis algorithms

In [9] we describe a combination of ideas to improve incremental signature-based Gröbner basis algorithms having a big impact on their performance. Besides explaining how to combine already known optimizations to achieve more efficient algorithms, we show how to improve them even more. Although our idea has a positive affect on all kinds of incremental signature-based algorithms, the way this impact is achieved can be quite different. Based on the two best-known algorithms in this area, F5 and G2V, we explain our idea, both from a theoretical and a practical point of view.

#### A new algorithmic scheme for computing characteristic sets

Ritt-Wu's algorithm of characteristic sets is the most representative for triangularizing sets of multivariate polynomials. Pseudo-division is the main operation used in this algorithm. In [18] we present a new algorithmic scheme for computing generalized characteristic sets by introducing other admissible reductions than pseudo-division. A concrete subalgorithm is designed to triangularize polynomial sets using selected admissible reductions and several effective elimination strategies and to replace the algorithm of basic sets (used in Ritt-Wu's algorithm). The proposed algorithm has been implemented and experimental results show that it performs better than Ritt-Wu's algorithm in terms of computing time and simplicity of output for a number of non-trivial test examples