Sparse Polynomial Interpolation Based on Derivative
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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 Bläser et al. for the polynomial given by an SLP over finite field (for large characteristic).
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