Related papers: The complexity and geometry of numerically solving…
This paper presents an alternative approach to simplify the proofs of some important results related to polynomial mappings in Computational Algebraic Geometry such as Polynomial Implicitization, Image Closure and some properties of the…
Reducing the conditions under which a given set satisfies the stipulations of the subset sum proposition to a set of linear relationships, the question of whether a set satisfies subset sum may be answered in a polynomial number of steps by…
In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation…
This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting…
Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…
We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…
Inspired by the work about solutions of a system of real polynomial equations done by Hermite, this paper introduces a Hermitian form, which encodes information about solutions of a system of complex polynomial equations with conjugate…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
Smale's alpha-theory uses estimates related to the convergence of Newton's method to give criteria implying that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements…
We describe an algorithm for fast multiplication of skew polynomials. It is based on fast modular multiplication of such skew polynomials, for which we give an algorithm relying on evaluation and interpolation on normal bases. Our…
In this article, we describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm…
The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k<x_1, ...,…
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…
Let $f$ be a polynomial system consisting of $n$ polynomials $f_1,\cdots, f_n$ in $n$ variables $x_1,\cdots, x_n$, with coefficients in $\mathbb{Q}$ and let $\langle f\rangle$ be the ideal generated by $f$. Such a polynomial system, which…
We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gr\"obner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation…
We use a generalization of Vinogradov's mean value theorem of S. Parsell, S. Prendiville and T. Wooley and ideas of W. Schmidt to give nontrivial bounds for the number of solutions to polynomial congruences, for arbitrary polynomials, when…
In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed…
Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to…