Section: New Results
Solving Systems over the Reals and Applications
Univariate real root isolation in an extension field and applications
In [11] we present algorithmic, complexity and implementation results for
the problem of isolating the real roots of a univariate polynomial
in
On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs
The number of embeddings of minimally rigid graphs in
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Towards this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, in [4] we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gröbner basis computations.
The Complexity of Subdivision for Diameter-Distance Tests
In [1] we present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization. Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight.
Real root finding for equivariant semi-algebraic systems
Let
Exact algorithms for semidefinite programs with degenerate feasible set
Let
A lower bound on the positive semidefinite rank of convex bodies
The positive semidefinite rank of a convex body
On the complexity of computing real radicals of polynomial systems
Let
Algorithms for Weighted Sums of Squares Decomposition of Non-negative Univariate Polynomials
It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a weighted sum of finitely many squares instead of a sum of two squares, then one can choose all coefficients in the representation to lie in the field generated by the coefficients of the polynomial. In particular, this allows an effective treatment of polynomials with rational coefficients. In [9], we describe, analyze and compare both from the theoretical and practical points of view, two algorithms computing such a weighted sums of squares decomposition for univariate polynomials with rational coefficients. The first algorithm, due to the third author relies on real root isolation, quadratic approximations of positive polynomials and square-free decomposition but its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm. They are exponential in the degree of the input univariate polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using quantifier elimination and root isolation bounds. The second algorithm, due to Chevillard, Harrison, Joldes and Lauter, relies on complex root isolation and square-free decomposition and has been introduced for certifying positiveness of poly-nomials in the context of computer arithmetics. Again, its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm, which are polynomial in the degree of the input polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using Vieta's formula and root isolation bounds. Finally, we report on our implementations of both algorithms and compare them in practice on several application benchmarks. While the second algorithm is, as expected from the complexity result, more efficient on most of examples, we exhibit families of non-negative polynomials for which the first algorithm is better.
On Exact Polya and Putinar's Representations
We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. In [18] we start by providing a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also compare the implementation of our algorithms with existing methods in computer algebra including cylindrical algebraic decomposition and critical point method.