In many cases, multivariate interpolation by smooth radial basis functions converges towards polynomial interpolants, when the basis functions are scaled to become “wide”. In particular, examples show that interpolation by scaled Gaussians seems to converge towards the de Boor/Ron “least” polynomial interpolant. The paper starts by providing sufficient criteria for the convergence of radial int...

The aim of this article is to introduce a generalization of the interpolation remainder in multivariate interpolation and to study it, in least interpolation schemes and in minimal interpolation ones. This generalization, we named λ-remainder, allows us a deeper analysis of the error in the interpolation process. Connected to the λ remainder, we introduced the notion of λ error order of interpo...

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the cla...

We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e. in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. Also a deterministic version of the algo...

In this paper we show that finding solutions of a system of multivariate polynomial equalities and inequalities in the max algebra is equivalent to solving an Extended Linear Complementarity Problem. This allows us to find all solutions of such a system of multivariate polynomial equalities and inequalities and provides a geometrical insight in the structure of the solution set. We also demonst...

The matrix eigenproblem approach for zero-dimensional multivariate polynomial system solving has been studied. The purpose of this paper is to apply the approach to a class of positive dimensional multivariate polynomial system which is called pseudo zero-dimensional system. In this case we can decide its reducibility, and when it is reducible, a set of polynomial can be determined from the eig...

Let K be a commutative ring and K be an affine space over K of dimension n. We introduce the concept of a family of multivariate maps f(n) of K into itself with invertible decomposition. If f(n) is computable in polynomial time then it can be used as the public rule and the invertible decomposition provides a private key in f(n) based public key infrastructure. Requirements of polynomial ity of...

We discuss four eigenvalue problems of increasing generality and complexity: rooting a univariate polynomial, solving the polynomial problem, set multivariate polynomials multi-parameter problems. In doing so, we provide unifying framework for these problems, where exploit properties (block-) (multi-) shift-invariant subspaces use multi-dimensional realization algorithms.

In this paper, we investigate property testing whether or not a degree d multivariate polynomial 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 uses random samples, nΩ(d) samples are required, which is not much fewer than it would take to completely determine the polynomial. To prove this lower bound, we ...