This article is an extended abstract of the ISSAC 2018 talk "Polynomial systems arising from discretizing systems of nonlinear differential equations" by Andrew Sommese.

This year we are celebrating the 10th anniversary of a dramatic revolution in combinatorial geometry, fueled by the infusion of techniques from algebraic geometry and algebra that have proven effective in solving a variety of hard problems that were ...

The first public beta release of GAP 4[6] was made on July 18 1997. Since then the system has been cited in over 2400 publications, and its distribution now includes over 130 contributed extension pack- ages. This tutorial will review the special ...

This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures.

Let A₀, ..., A_n be m x m symmetric matrices with entries in Q, and let A(x) be the linear pencil A₀+x₁ A₁ + ··· + x_n A_n, where x=(x₁,...,x_n) are unknowns. The linear matrix inequality (LMI) A(x) ≥ 0 defines the subset of Rⁿ, called spectrahedron, ...

One of the biggest open problems in computational algebra is the design of efficient algorithms for Gröbner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed ...

In this paper, we propose algorithms for computing minimal associated primes of ideals in polynomial rings over Q and computing radicals of ideals in polynomial rings over a field. They apply Chinese Remainder Theorem (CRT) to Laplagne's algorithm which ...

Many special functions as well as generating functions of combinatorial sequences that arise in applications are D-finite, i.e., they satisfy a linear differential equation with polynomial coefficients. These functions have been studied for centuries ...

We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows the framework developed by Duff et al. (2018) and can operate in the ...

We provide a complete enumeration of all complex Golay pairs of length up to 25, verifying that complex Golay pairs do not exist in lengths 23 and 25 but do exist in length 24. This independently verifies work done by F. Fiedler in 2013 that confirms ...

Let L be a 4th order linear differential operator with coefficients in K(z), with K a computable algebraically closed field. The operator L is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions X satisfies X^t ...

Let G be the reduced Grö bner basis of a zero-dimensional ideal I ⊆ K[X, Y] of bivariate polynomials over an effective field K. Modulo suitable regularity assumptions on G and suitable precomputations as a function of G , we prove the existence of a ...

We provide a new framework for a posteriori validation of vector-valued problems with componentwise tight error enclosures, and use it to design a symbolic-numeric Newton-like validation algorithm for Chebyshev approximate solutions of coupled systems ...

An algorithm for interpolating a polynomial f from evaluation points whose running time depends on the sparsity t of the polynomial when it is represented as a sum of t Chebyshev Polynomials of the First Kind with non-zero scalar coefficients is given ...

In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P ın F_q [x] of degree <n at all n -th roots of unity in F_q^d can essentially be computed d times faster than evaluating Q ın F_q^d x at all these roots, assuming F_q^d contains a ...

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist - sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it ...

The number of embeddings of minimally rigid graphs in R^D is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower ...

For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal approximant ...

We consider the computation of two normal forms for matrices over the univariate polynomials: the Popov form and the Hermite form. For matrices which are square and nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we ...

We show that, for cyclic convolutional codes, it is possible to compute a sequence of positive integers, called cyclic column distances, which presents a more regular behavior than the classical column distances sequence. We then design an algorithm for ...