Related papers: Optimal Points for Cubature Rules and Polynomial I…

In this paper, we investigate property testing whether or not a degree d multivariate poly- nomial is a sum of squares or is far from a sum of squares. We show that if we require that the property tester always accepts YES instances and…

We consider fractional Sobolev spaces $H^\theta$, $\theta\in (0,1)$, on 2D domains and $H^1$-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shape-regular triangles and quadrilaterals. We…

This paper provides approximation orders for a class of nonlinear interpolation procedures for univariate data sampled over $\sigma$ quasi-uniform grids. The considered interpolation is built using both essentially nonoscillatory (ENO) and…

This paper is concerned with achieving optimal coherence for highly redundant real unit-norm frames. As the redundancy grows, the number of vectors in the frame becomes too large to admit equiangular arrangements. In this case, other…

Let $f_1,\dots,f_m$ be polynomials in $n$ variables with coefficients in a finite field $\mathbb{F}_q$. We estimate the number of points $\underline{x}$ in $\mathbb{F}_q^n$ such that each value $f_i(\underline{x})$ is a nonzero square in…

Scalar curvature constraints can be studied by means of splitting procedures. The success of this strategy depends on the control we can get on its splitting factors. We introduce canonical so-called minimal splitting factors. They have…

We study the error in quadrature rules on a compact manifold. As in the Koksma-Hlawka inequality, we consider a discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative…

Given a polynomial system f associated with a simple multiple zero x of multiplicity {\mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited…

Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for…

It is well known that the bivariate polynomial interpolation problem at domain points of a triangle is correct. Thus the corresponding interpolation matrix $M$ is nonsingular. L.L. Schumaker stated the conjecture, that the determinant of…

We study minimal reductions of edge ideals of graphs and determine restrictions on the coefficients of the generators of these minimal reductions. We prove that when $I$ is not basic, then $\core{I}\subset \m I$, where $I$ is an edge ideal…

In this paper, we investigate the problem of designing compact support interpolation kernels for a given class of signals. By using calculus of variations, we simplify the optimization problem from an infinite nonlinear problem to a finite…

We introduce a class of ideals generated by a set of 2-minors of $m\times n$-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by…

This paper applies a complete parametric set for approximating the geometry of a quadrilateral element. The approximation basis used is a complete Pascal polynomial of second order with six free parameters. The interpolation procedure is a…

An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional…

Let $M$ be an open Riemann surface and let $\Lambda\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $\Lambda$ and…

While it is a classical result dating back to Dehn (1903) that squares composing a perfect rectangle must have rational side lengths, the arithmetic complexity of these tilings, specifically the growth of the denominators of these rational…

We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the…

We show that the reduced point variety of a quantum polynomial algebra is the union of specific linear subspaces in $\mathbb{P}^n$, we describe its irreducible components and give a combinatorial description of the possible configurations…

A new alternative numerical procedure to the Szeg\H{o} quadrature formulas for the estimation of integrals with respect to a positive Borel measure $\mu$ supported on the unit circle is presented. As in many practical situations, we assume…