We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the algorithm of BenOr and Tiwari from 1988 for interpolating polynomials over rings with characteristic zero to characteristic p by doing additional probes.To interpolate a polynomial in n variables with t non-zero terms, Zippel's (1990) algorithm interpolates one variable at a time using O(ndt) probes to the black box where d bounds the degree of the polynomial. Our new algorithm does O(nt) probes. It interpolates each variable independently using O(t) probes which allows us to parallelize the main loop giving an advantage over Zippel's algorithm.We have implemented both Zippel's algorithm and the new algorithm in C and we have done a parallel implementation of our algorithm using Cilk [2]. In the paper we provide benchmarks comparing the number of probes our algorithm does with both Zippel's algorithm and Kaltofen and Lee's hybrid of the Zippel and Ben-Or/Tiwari algorithms.
We present a first sparse modular algorithm for computing a greatest common divisor of two polynomials f1, f2where L is an algebraic function field in k ≥ 0 parameters with r ≥ 0 field extensions. Our algorithm extends the dense algorithm of Monagan and van Hoeij from 2004 to support multiple field extensions and to be efficient when the gcd is sparse. It uses the modified Zippel interpolation of de Kleine, Monagan, and Wittkopf from 2005.We have implemented our algorithm in Maple. We provide timings demonstrating the efficiency of our algorithm compared to that of Monagan and van Hoeij and with a primitive fraction-free Euclidean algorithm for both dense and sparse gcd problems.
We present an efficient algorithm for factoring a multivariate polynomial f ∈ L[x1, . . . , xv] where L is an algebraic function field with k ≥ 0 parameters t1, . . . , t k and r ≥ 0 field extensions. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over Q. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization of f when using Hensel lifting.We have implemented our algorithm in Maple 13. We provide timings demonstrating the efficiency of our algorithm.
Extended Abstract of Poster Presented at ISSAC 2010Let f be a polynomial in n variables x 1 , x 2 , ..., x n of degree d with t non-zero terms over a finite field with characteristic p. It is well known that f can be interpolated from (1 + d) n points using Newton interpolation. But this is exponential in n. If t(1 + d) n , that is, f is sparse, we seek algorithms for interpolating f whose complexity is polynomial in d, t and n.The first such algorithm was engineered by Richard Zippel in 1979 (see [9]). Zippel's algorithm interpolates one variable at a time using O(ndt) points. Zippel's algorithm is probabilistic; it needs p ndt to succeed with high probability. Zippel developed his algorithm to interpolate a sparse polynomial GCD with integer coefficients. Zippel's algorithm is now used in Maple, Magma and Mathematica as the default GCD algorithm for multivariate GCD computation over the integers. Rayes, Wang and Weber attempted a parallel implementation in [8]. In [4] we extended it to compute GCDs over algebraic function fields.The second such algorithm was developed by Ben-Or and Tiwari in 1988 for interpolation in characteristic 0.
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