Related papers: Real-rooted integer polynomial enumeration algorit…
We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Sch{\"o}nhage's efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but…
Local search has recently been applied to SMT problems over various arithmetic theories. Among these, nonlinear real arithmetic poses special challenges due to its uncountable solution space and potential need to solve higher-degree…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of \emph{Kadison-Singer} , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a…
We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are…
Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous…
We study the theoretical and practical aspects of computing braids described by approximate descriptions of paths in the plane. Exact algorithms rely on the lexicographic ordering of the points in the plane, which is unstable under…
Fast exact algorithms are known for Hamiltonian paths in undirected and directed bipartite graphs through elegant though involved algorithms that are quite different from each other. We devise algorithms that are simple and similar to each…
We study the existence of polynomial kernels for the problem of deciding feasibility of integer linear programs (ILPs), and for finding good solutions for covering and packing ILPs. Our main results are as follows: First, we show that the…
In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…
Evaluating a polynomial on a set of points is a fundamental task in computer algebra. In this work, we revisit a particular variant called trimmed multipoint evaluation: given an $n$-variate polynomial with bounded individual degree $d$ and…
We present an algorithm for computing asymptotic approximations of roots of polynomials with exp-log function coefficients. The real and imaginary parts of the approximations are given as explicit exp-log expressions. We provide a method…
This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…
Block full rank pencils introduced in [Dopico et al., Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…
In this paper, we introduce a new family of codes relevent for rank and sum-rank metrics. These codes are based on an effective Chinese remainders theorem for linearized polynomials over finite fields. We propose a decoding algorithm for…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We prove that any lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This has consequences for problems in several areas of mathematics. We use it to develop a general theory for…
In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this…
Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…