Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems

B{\'e}zout 's theorem states that dense generic systems of n multivariate quadratic equations in n variables have 2 n solutions over algebraically closed fields. When only a small subset M of monomials appear in the equations (fewnomial systems), the number of solutions may decrease dramatically. We focus in this work on subsets of quadratic monomials M such that generic systems with support M do not admit any solution at all. For these systems, Hilbert's Nullstellensatz ensures the existence of algebraic certificates of inconsistency. However, up to our knowledge all known bounds on the sizes of such certificates -including those which take into account the Newton polytopes of the polynomials- are exponential in n. Our main results show that if the inequality 2|M| -- 2n $\le$ \sqrt 1 + 8{\nu} -- 1 holds for a quadratic fewnomial system -- where {\nu} is the matching number of a graph associated with M, and |M| is the cardinality of M -- then there exists generically a certificate of inconsistency of linear size (measured as the number of coefficients in the ground field K). Moreover this certificate can be computed within a polynomial number of arithmetic operations. Next, we evaluate how often this inequality holds, and we give evidence that the probability that the inequality is satisfied depends strongly on the number of squares. More precisely, we show that if M is picked uniformly at random among the subsets of n + k + 1 quadratic monomials containing at least $\Omega$(n 1/2+$\epsilon$) squares, then the probability that the inequality holds tends to 1 as n grows. Interestingly, this phenomenon is related with the matching number of random graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results showing that certificates in inconsistency can be computed for systems with more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201

Change of basis for m-primary ideals in one and two variables

Following recent work by van der Hoeven and Lecerf (ISSAC 2017), we discuss the complexity of linear mappings, called untangling and tangling by those authors, that arise in the context of computations with univariate polynomials. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings, and use them to give bounds on the arithmetic complexity of certain algebras.Comment: In Proceedings ISSAC'19, ACM, New York, USA. See proceedings version for final formattin

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use $\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ operations in $\mathbb{K}$, where $s$ is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and $\omega$ is the exponent of matrix multiplication. The soft-$O$ notation indicates that logarithmic factors in the big-$O$ are omitted while the ceiling function indicates that the cost is $\widetilde{\mathcal{O}}(n^\omega)$ when $s = o(1)$. Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm

Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pad\'e approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between $m$ vectors of size $\sigma$; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ field operations, where $\omega$ is the exponent of matrix multiplication and the notation $\mathcal{O}\tilde{~}(\cdot)$ indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pad\'e approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size $\Theta(m^2 \sigma)$, the cost bound $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always $\mathcal{O}(m \sigma)$. Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms

A SAT+CAS Approach to Finding Good Matrices: New Examples and Counterexamples

We enumerate all circulant good matrices with odd orders divisible by 3 up to order 70. As a consequence of this we find a previously overlooked set of good matrices of order 27 and a new set of good matrices of order 57. We also find that circulant good matrices do not exist in the orders 51, 63, and 69, thereby finding three new counterexamples to the conjecture that such matrices exist in all odd orders. Additionally, we prove a new relationship between the entries of good matrices and exploit this relationship in our enumeration algorithm. Our method applies the SAT+CAS paradigm of combining computer algebra functionality with modern SAT solvers to efficiently search large spaces which are specified by both algebraic and logical constraints

Constructing minimal telescopers for rational functions in three discrete variables

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach