Related papers: Observations on interpolation by total degree poly…

The paper contains a survey of the results obtained by the author in recent years. These results concern the application in multivariate polynomial interpolation of some geometric constructions and methods. In particular, we give estimates…

One frequently needs to interpolate or approximate gradients on simplicial meshes. Unfortunately, there are very few explicit mathematical results governing the interpolation or approximation of vector-valued functions on Delaunay meshes in…

We present certain existence criteria and parameterisations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to…

We consider interpolation of univariate functions on arbitrary sets of nodes by Gaussian radial basis functions or by exponential functions. We derive closed-form expressions for the interpolation error based on the…

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as…

It is well known, that if polynomial with rational coefficients of degree $n$ takes integer values in points $0,1,...,n$ then it takes integer values in all integer points. Are there sets of $n+1$ points with the same property in other…

In this paper we characterize, in algebraic and geometric terms, when a graded vanishing ideal is generated by binomials over any field K.

The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Pad\'e approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ,…

We show that a global holomorphic section of $\mathscr{O}(d)$ restricted to a closed complex subspace $X \subset \mathbb{P}^n$ has an interpolant if and only if it satisfies a set of moment conditions that involves a residue current…

This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial…

Probabilistic completeness is an important property in motion planning. Although it has been established with clear assumptions for geometric planners, the panorama of completeness results for kinodynamic planners is still incomplete, as…

Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding…

In this paper we consider monomial localizations of monomial ideals and conjecture that a monomial ideal is polymatroidal if and only if all its monomial localizations have a linear resolution. The conjecture is proved for squarefree…

In a recent paper by Harada, Seceleanu, and \c{S}ega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is…

We describe an effective method for calculating certain infinite sums, generalizations of the classical Bernoulli polynomials. As shown by Edward Witten in his papers on two-dimensional gauge theories, the correlation functions of…

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…

We study the bifurcation values of real polynomial maps $f: \bR^{2n} \to \bR^2$ which reflect the lack of asymptotic regularity at infinity. We formulate real counterparts of some structure results which have been previously proved in case…

High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting…

We study the commutative algebra of three bihomogeneous polynomials p_0,p_1,p_2 of degree (2,1) in variables x,y;z,w, assuming that they never vanish simultaneously on P^1 x P^1. Unlike the situation for P^2, the Koszul complex of the p_i…

We construct a Gr\"obner Basis of the relation ideal of a polynomial, give an interpolation formula for the basis elements and explain the connection of the interpolation formula to the Buchberger--M\"oller algorithm. We present a situation…