Related papers: Deciding Nonnegativity of Polynomials by MAPLE
A linear polyomial non-negative on the non-negativity domain of finitely many linear polynomials can be expressed as their non-negative linear combination. Recently, under several additional assumptions, Helton, Klep, and McCullough…
Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous…
We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition. We follow the approach of untrusted certificates,…
We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which…
Our focus is upon {\it irreducible} nonnegative $n$-by-$n$ matrix realizations of nonnegatively realizable spectra or, equivalently, characteristic polynomials. After giving some general background, we make some useful new observations and…
Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…
Let L be a restricted Lie superalgebra with its restricted enveloping algebra u(L) over a field F of characteristic p>2. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2\times 2 matrices over F. We…
A set $S\subset \mathbb{R}^n$ is a nonnegativity witness for a set $U$ of real homogeneous polynomials if $F$ in $U$ is nonnegative on $\mathbb{R}^n$ if and only if it is nonnegative at all points of $S$. We prove that the union of the…
In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the…
This paper presents a fast and effective computer algebraic method for analyzing and verifying non-linear integer arithmetic circuits using a novel algebraic spectral model. It introduces a concept of algebraic spectrum, a numerical form of…
We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…
Various methods can obtain certified estimates for roots of polynomials. Many applications in science and engineering additionally utilize the value of functions evaluated at roots. For example, critical values are obtained by evaluating an…
Student appreciation of a function is enhanced by understanding the graphical representation of that function. From the real graph of a polynomial, students can identify real-valued solutions to polynomial equations that correspond to the…
An invaluable feature of computer algebra systems is their ability to plot the graph of functions. Unfortunately, when one is trying to design a library of mathematical functions, this feature often falls short, producing incorrect and…
Certifying nonnegativity of polynomials is a well-known NP-hard problem with direct applications spanning non-convex optimization, control, robotics, and beyond. A sufficient condition for nonnegativity is the Sum of Squares (SOS) property,…
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 present MultivariatePowerSeries, a Maple library introduced in Maple 2021, providing a variety of methods to study formal multivariate power series and univariate polynomials over such series. This library offers a simple and easy-to-use…
A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
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…