Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution

In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite field, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-or and Tiwarri algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields

Sparse Polynomial Interpolation by Variable Shift in the Presence of Noise and Outliers in the Evaluations

We compute approximate sparse polynomial models of the form ˜ f(x) = ∑t ej j=1 ˜cj(x−˜s) to a function f(x), of which an approximation of the evaluation f(ζ) at any complex argument value ζ can be obtained. We assume that several of the returned function evaluations f(ζ) are perturbed not just by approximation/noise errors but also by highly perturbed outliers. None of the ˜cj, ˜s, ej and the location of the outliers are known before-hand. We use a numerical version of an exact algorithm by [Giesbrecht, Kaltofen, and Lee 2003] together with a numerical version of the Reed-Solomon error correcting coding algorithm. We also compare with a simpler approach based on root finding of derivatives, while restricted to characteristic 0. In this preliminary report, we discuss how some of the problems of numerical instability and ill-conditioning in the arising optimization problems can be overcome. By way of experiments, we show that our techniques can recover approximate sparse shifted polynomial models, provided that there are few terms t, few outliers and that the sparse shift is relatively small

Symbolic Recipes for Polynomial System Solving

Contribution à un ouvrage.International audienceIn many branches of science and engineering where mathematics is used, the resolution of a problem coming from practice is often reduced to the search of a solution for a system of (algebraic or differential) equations modelling the considered problem. From our point of view, to solve a polynomial system of equations is to rewrite it (i.e., to present it in a different form) in such a way that some ‘nontrivial’ information about its solutions can be derived from this new presentation. The information mentioned above can be related to the existence or non-existence of complex or real solutions, to the number of real or complex solutions, to the approximation of one or several solutions, etc