Accelerated Solution of Multivariate Polynomial Systems of Equations

We propose new Las Vegas randomized algorithms for the solution of a square nondegenerate system of equations, with well-separated roots. The algorithms use $\Oc (\delta\, \csttn D^{2} \log(D) \log(b))$ arithmetic operations (in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal) 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 complex roots of the system (e.g., Bezout or Bernshtein bound), $\delta$ is the number of real roots or the roots lying in the box or disc, and $\epsilon=2^{-b}$ is the required upper bound on the output errors. For computing the normal form modulo the ideal, the efficient practical algorithms of [B. Mourrain and P. Trebuchet, in Proceedings of the International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2000, pp. 231--238] or [J. C. Faugere, J. Pure Appl. Algebra, 139 (1999), pp. 61--88] can be applied. We also yield the bound $\Oc( \csttn D^{2} \log(D) )$ on the complexity of counting the numbers of all roots in a fixed box (disc) and all real roots. For a large class of inputs and typically in practical computations, the factor $\delta$ is much smaller than $D, \delta=o(D)$. This improves by the order of magnitude the known complexity estimates of the order of at least 3n D4 + D3 log(b) or D4, which so far are the record estimates even for the approximation of 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 normal forms of polynomials in the quotient algebra; 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.