Related papers: Coamoebas of polynomials supported on circuits
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.…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for…
Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this…
In this work we establish a connection between copositivity, that is, nonnegativity on the positive orthant, of sparse real Laurent polynomials and discriminants. Specifically, we consider Laurent polynomials in the positive orthant with…
We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
In recent years, finding new satisfiability algorithms for various circuit classes has been a very active line of research. Despite considerable progress, we are still far away from a definite answer on which circuit classes allow fast…
Explicit monoid structure is provided for the class of canonical subfield preserving polynomials over finite fields. Some classical results and asymptotic estimates will follow as corollaries.
We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…
We propose an algebraic approach to proving circuit lower bounds for ACC0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC0 and ACC0 can be reformulated in…
We primarily investigate the properties of characteristic polynomials of semimatroids. In particular, we provide a combinatorial interpretation of their coefficients, generalizing the Whitney's Broken Circuit Theorem. We also prove that the…
We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…
It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification…
Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for…
We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…
This paper deals with coamoebas, that is, images under coordinatewise argument mappings, of certain quite particular plane algebraic curves. These curves are the zero sets of reduced A-discriminants of two variables. We consider the…
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots)…
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…