Related papers: Refining Multivariate Value Set Bounds
Letting $P$ be a convex polytope in $\mathbb{R}^d$ with $n>d$ vertices, we study geometric and analytical properties of the set of generalized barycentric coordinates relative to any point $p\in P$. We prove that such sets are polytopes in…
A short note on bounds on distance to variety of a point in terms of the Taylor coefficients at the point.
We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface…
A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial,…
Let (K, v) be a henselian valued field of arbitrary rank. In this paper, we give an irreducibility criterion for multivariate polynomials over K using valuation theory.
In this paper, we investigate geometrical properties of the rank metric space and covering properties of rank metric codes. We first establish an analytical expression for the intersection of two balls with rank radii, and then derive an…
Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions…
In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion's formula for the…
We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical…
We establish a version of the Pommerenke-Levin-Yoccoz inequality for the modulus of a polynomial-like restriction of a global polynomial and give two applications. First it is shown that if the modulus of a polynomial-like restriction of an…
Given a bivariate system of polynomial equations with fixed support sets $A, B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the…
The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state…
We consider the problem of deciding the satisfiability of quantifier-free formulas in the theory of finite sets with cardinality constraints. Sets are a common high-level data structure used in programming; thus, such a theory is useful for…
The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Groebner bases methods. In this paper, we consider polynomial systems that are…
In this work we investigate unions of lifted MRD codes of a fixed dimension and minimum distance and derive an explicit formula for the cardinality of such codes. This will then imply a lower bound on the cardinality of constant dimension…
A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by…
We show that the set of complex points in the moduli space of polynomials of degree d corresponding to post-critically finite polynomials is a set of algebraic points of bounded height. It follows that for any B, the set of conjugacy…
Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than $ 1 $ are derived. An application in coding theory is illustrated by showing that multispace…
We study degree bounds on rational but not necessarily polynomial generators for the field $\mathbf{k}(V)^G$ of rational invariants of a linear action of a finite abelian group. We show that lattice-theoretic methods used recently by the…
This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…