Related papers: The multivariate Hermite method for counting real …
In this paper we consider finite-dimensional constrained Hamiltonian systems of polynomial type. In order to compute the complete set of constraints and separate them into the first and second classes we apply the modern algorithmic methods…
This work presents a framework for a-posteriori error-estimating algorithms for differential equations which combines the radii polynomial approach with Haar wavelets. By using Haar wavelets, we obtain recursive structures for the matrix…
We introduce a Macaulay2 package for working with jet schemes. The main method constructs jets of ideals, polynomial rings and their quotients, ring homomorphisms, affine varieties, and (hyper)graphs. The package also includes additional…
In this work, we introduce a novel Hermite method to handle Maxwell's equations for nonlinear dispersive media. The proposed method achieves high-order accuracy and is free of any nonlinear algebraic solver, requiring solving instead small…
The Hermite polynomials are ubiquitous but can be difficult to work with due to their unwieldy definition in terms of derivatives. To remedy this, we showcase an underappreciated Gaussian integral formula for the Hermite polynomials, which…
The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive'…
For bivariate polynomials of degree $n\le 5$ we give fast numerical constructions of determinantal representations with $n\times n$ matrices. Unlike some other available constructions, our approach returns matrices of the smallest possible…
We study the computational complexity of a diagonalization technique for multivariate homogeneous polynomials, that is, expressing them as sums of powers of independent linear forms. It is based on Harrison's center theory and consists of a…
Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy…
For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…
Bayesian variable selection is a powerful tool for data analysis, as it offers a principled method for variable selection that accounts for prior information and uncertainty. However, wider adoption of Bayesian variable selection has been…
This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a…
Finding the solutions to a system of multivariate polynomial equations is a fundamental problem in mathematics and computer science. It involves evaluating the polynomials at many points, often chosen from a grid. In most current methods,…
We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
The aim of the paper is to present Hermite-type multiwavelets satisfying the vanishing moment property with respect to elements in the space spanned by exponentials and polynomials. Such functions satisfy a two-scale relation which is…
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…
In this paper some algorithms will be presented which can be used for the calculation of zeros of polynomials and eigenvalues of polynomial matrices with a multiplicity larger than one. The numerical values calculated with MATLAB are used…
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…
We present an effective criterion for determining whether a (augmented) vertically parametrized polynomial system admits multiple positive zeros for some choice of parameter values. Our method builds on previous algorithms from chemical…