Sparse Polynomial Interpolation Based on Derivative

Abstract: In this paper, we propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP). Both of our algorithms work over any finite fields $F_q$ with large characteristic. The first one is a Monte Carlo randomized algorithm. Its arithmetic complexity is linear in the number $T$ of non-zero terms of $f$, in the number $n$ of variables. If $q$ is $O((nTD)^{(1)})$, where $D$ is the partial degree bound, then our algorithm has better complexity than other existing algorithms. The second one is a deterministic algorithm. It has better complexity than existing deterministic algorithms over a field with large characteristic. Its arithmetic complexity is quadratic in $n,T,\log D$, i.e., quadratic in the size of the sparse representation. And we also show that the complexity of our deterministic algorithm is the same as the one of deterministic zero-testing of Bl\"{a}ser et al. for the polynomial given by an SLP over finite field (for large characteristic).