MutantXL is an algorithm for solving systems of polynomial equations that was proposed at SCC 2008 and improved in PQC 2008. This article gives an overview over the MutantXL algorithm. It also presents experimental results comparing the behavior of the MutantXL algorithm to the F4 algorithm on HFE and randomly generated multivariate systems. In both cases MutantXL is faster and uses less memory...

A new algorithm for factoring multivariate polynomials over the integers based on an algorithm by Wang and Rothschild is described. The new algorithm has improved strategies for dealing with the known problems of the original algorithm, namely, the leading coefficient problem, the bad-zero problem and the occurrence of extraneous factors. It has an algorithm for correctly predetermining leading...

Abstract: We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. The geometry of the design is recovered via semidefinite programming duality theory. Th...

Multivariate polynomial interpolation is a basic and fundamental subject in Approximation Theory and Numerical Analysis, which has received and continues receiving not deep but constant attention. In this short survey we review its development in the rst 75 years of this century, including a pioneering paper by Kronecker in the 19th century. x1. Introduction Interpolation, by polynomials or oth...

In this paper we study multivariate polynomial interpolation on lower sets of points. A lower set can be expressed as the union of blocks of points. We show that a natural interpolant on a lower set can be expressed as a linear combination of tensor-product interpolants over various intersections of the blocks that define it. Math Subject Classification: 41A05, 41A10, 65D05

We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to Ben-Or and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz systems arising in the Ben-Or and Tiwari algorithm and for solving transposed Vandermonde systems of...

An enclosure is a two-sided approximation of a unior multivariate function by a pair of typically simpler functions such that over the domain of interest. Enclosures are optimized by minimizing the width and refined by enlarging the space . This paper develops a framework for efficiently computing enclosures for multivariate polynomials and, in particular, derives piecewise bilinear enclosures ...

We discuss three natural pseudodistances and pseudometrics on a bounded domain in IR based on polynomial inequalities.

It is often the case that the exact moments of a continuous distribution whose support is confined to a closed interval can be explicitly determined, while its exact density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. A representation of a least-squares approximating polynomial which does not rely on orthogonal polynomials, is obtaine...

New Approximations of the Total Variation and Filters in Image Processing