Improved Hermite multivariate polynomial interpolation

Abstract

In this paper we give an algorithm with complexity O`µ 2´t o solve Hermite multivariate polynomial interpolation with µ conditions on its Hasse derivatives. In the case of bivariate interpolation used to perform list-decoding on Reed-Solomon of length n and dimension k with multiplicity m on each point, it permits to obtain a complexity in O`n 2 m 4´w hich does not depend on the rate k/n and better than previously known complexity in O " n 2 m 5 (n/k). This algorithm can also be used for recent interpolation list-decoding with three and more variables. For interpolation on polynomial with n points and M variables with prescribed multiplication order m the general complexity of the algorithm is O`n 2 m 2M´.

Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations

“…This approach based on the computation of a reduced lattice basis was in particular the basis of the extensions to the multivariate case s > 1 in [10,9,14]. In the multivariate case as well, the result in Theorem 1 improves on the best previously known bounds [10,9,14]; we detail those bounds and we prove this claim in Appendix C. In [18], the authors solve a problem similar to Problem 1 except that they do not assume that the x i are distinct. For simple roots and under some genericity assumption on the points {(x i , y i,1 , .…”