Related papers: Solving sparse polynomial systems using Groebner b…
Two complementary approaches have been extensively used in signal and image processing leading to novel results, the sparse representation methodology and the variational strategy. Recently, a new sparsity based model has been proposed, the…
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…
We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface…
What can be (machine) learned about the complexity of Buchberger's algorithm? Given a system of polynomials, Buchberger's algorithm computes a Gr\"obner basis of the ideal these polynomials generate using an iterative procedure based on…
Deep Neural Networks are powerful tools for solving machine learning problems, but their training often involves dense and costly parameter updates. In this work, we use a novel Max-Plus neural architecture in which classical addition and…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…
We present a new algorithm for computing a truncated Markov basis of a lattice. In general, this new algorithm is faster than existing methods. We then extend this new algorithm so that it solves the linear integer feasibility problem with…
We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Signature-based algorithms are a popular kind of algorithms for computing Groebner basis, including the famous F5 algorithm, F5C, extended F5, G2V and the GVW algorithm. In this paper, an efficient method is proposed to solve the…
The Bernstein polynomial basis sees significant use owing to its unique properties, particularly in the field of optimal control. However, the basis is known to have a slow rate of convergence to the function it approximates. With this in…
Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing…
This article is partly a survey and partly a research paper. It tackles the use of Groebner bases for addressing problems of numerical semigroups, which is a topic that has been around for some years, but it does it in a systematic way…
In this paper we present an algorithm for computing Groebner bases of linear ideals in a difference polynomial ring over a ground difference field. The input difference polynomials generating the ideal are also assumed to be linear. The…
We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of…
We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree…
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian which gave, together with G.Singh, an algorithm to compute their Groebner bases. Although their algorithm has polynomial complexity when the chain modules are…
Given a finite set of closed rational points of affine space over a field, we give a Gr\"obner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the…