Complexity of Resolution of Parametric Systems of Polynomial Equations and Inequations

TL;DR

This paper presents a deterministic method for computing the minimal discriminant variety of parametric polynomial systems, bounding its degree and analyzing the complexity of the approach, which aids in understanding solution behavior over parameters.

Contribution

It introduces a new deterministic elimination-based method for computing the minimal discriminant variety with proven degree bounds and complexity analysis for systems with parameters.

Findings

Degree of minimal discriminant variety bounded by D:=(n+r)d^{(n+1)}

Method complexity is σ^{O(1)} D^{O(n+s)} bit-operations

Applicable to systems with radical zero-dimensional ideals in generic cases

Abstract

Consider a system of n polynomial equations and r polynomial inequations in n indeterminates of degree bounded by d with coefficients in a polynomial ring of s parameters with rational coefficients of bit-size at most $\sigma$. From the real viewpoint, solving such a system often means describing some semi-algebraic sets in the parameter space over which the number of real solutions of the considered parametric system is constant. Following the works of Lazard and Rouillier, this can be done by the computation of a discriminant variety. In this report we focus on the case where for a generic specialization of the parameters the system of equations generates a radical zero-dimensional ideal, which is usual in the applications. In this case, we provide a deterministic method computing the minimal discriminant variety reducing the problem to a problem of elimination. Moreover, we prove that the degree of the computed minimal discriminant variety is bounded by $D:=(n+r)d^{(n+1)}$ and that the complexity of our method is $\sigma^{\mathcal{O}(1)} D^{\mathcal{O}(n+s)}$ bit-operations on a deterministic Turing machine.