Topics in Comptuter Algebra: Lecture 3
Sparse polynomial interpolation and polynomial GCD computation
Michael Monagan, Department of Mathematics, Simon Fraser University.
Abstract: We will present Brown's dense modular GCD algorithm from 1971 and Zippel's sparse modular GCD algorithm from 1979. For a prime p, Brown's algorithm recursively interpolates the gcd G of A and B in Zp[x1,...xn] from images in in one less variable using a dense interpolation. If the gcd G is sparse in many variables, e.g., G = x1^d + x2^d + ... + xn^n + 1, this interpolation requires at least (n-1)^(d+1) univariate images in Zp[x1], which is exponential in n, the number of variables. In comparison, Zippel's sparse interpolation reduces this to O(n^2d). Zippel's algorithm is one of the first probabilistic algorithms. A variation on it is now using in Maple 11.