Asymptotic acceleration of solving multivariate polynomial systems of equations

Abstract

We propose new Las Vegas randomized algorithms for the solution of a multivariate generic or sparse polynomial system of equations. The algorithms useO (( +4n)3nD2 log b) arithmetic operations to approximate all real roots of the system as well as all roots lying in a fixed n-dimensional box or disc. Here D is an upper bound on the number of all the roots of the system, is the number of real roots or the roots lying in the box or disc, = 2 b is the required upper bound on the output errors, and O (s) stands for O(s logc s), c being a constant independent of s. We also yield the bounds O (12nD2) for the complexity of counting the numbers of all roots in a fixed box (disc) and all real roots and O (12nD2 log b) for the complete solution of generic system. For a large class of inputs and typically in practical computations, the factor is much smaller than D; = o(D). This improves by order of magnitude the known complexity estimates of order at least D3 log b or D3, which so far are the record ones even for approximating a single root of a system and for each of the cited counting problems, respectively. Our progress relies on proposing several novel techniques. In particular, we exploit the structure of matrices associated to a given polynomial system and relate it to the associated linear operators, dual space of linear forms, and algebraic residues; furthermore, our techniques support the new nontrivial extension of the matrix sign and quadratic inverse power iterations to the case of multivariate polynomial systems, where we emulate the recursive splitting of a univariate polynomial into factors of smaller degree.