Related papers: Extensions and Corrections for: ``A Convex Geometr…

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete…

In this paper, we provide a new method to find all zeros of polynomials with quaternionic coefficients located on only one side of the powers of the variable (these polynomials are called simple polynomials). This method is much more…

We present an elementary proof of a conjecture proposed by I. Rasa in 2017 which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive by A. Komisarski and T. Rajba very recently by the use…

Work in progress concerning alternative formalizations of arithmetic.

The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by…

The notion of root polynomials of a polynomial matrix $P(\lambda)$ was thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020]. In this…

We apply techniques from symplectic geometry to extend and give a new proof of the complex convexity theorem of Gindikin-Kroetz.

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive…

In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based…

We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to…

This paper derives numerical bounds for and implements the splitting circle method for finding roots of a univariate polynomial in the presence of fixed precision.

Comment on ``Gibbs Sampling, Exponential Families and Orthogonal Polynomials'' [arXiv:0808.3852]

We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…

This part is made of three sections. In the first section we study the family of polynomials whose roots are 4cos2 k{\pi}, (n \geqslant 3,1 \leqslant k < \frac{n}{2}). We obtain n2 in this manner a family of orthogonal polynomials. This…

Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution…

Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of…

This note generalizes the Fibonacci primitive roots to the set of integers. An asymptotic formula for counting the number of integers with such primitive root is introduced here.