Polynomial-time computing over quadratic maps I: sampling in real algebraic sets

TL;DR

This paper introduces a polynomial-time algorithm for sampling points in real algebraic sets defined by quadratic maps and polynomial compositions, significantly improving efficiency when the target dimension is small.

Contribution

It presents a novel, efficient procedure for sampling in real algebraic sets with complexity polynomial in key parameters, surpassing previous exponential-time methods.

Findings

Algorithm computes sampling points in polynomial time for fixed k.

Provides bounds on the number of connected components of the algebraic set.

Extends real algebraic set analysis using infinitesimals and limit computation.

Abstract

Given a quadratic map Q : K^n -> K^k defined over a computable subring D of a real closed field K, and a polynomial p(Y_1,...,Y_k) of degree d, we consider the zero set Z=Z(p(Q(X)),K^n) of the polynomial p(Q(X_1,...,X_n)). We present a procedure that computes, in (dn)^O(k) arithmetic operations in D, a set S of (real univariate representations of) sampling points in K^n that intersects nontrivially each connected component of Z. As soon as k=o(n), this is faster than the standard methods that all have exponential dependence on n in the complexity. In particular, our procedure is polynomial-time for constant k. In contrast, the best previously known procedure (due to A.Barvinok) is only capable of deciding in n^O(k^2) operations the nonemptiness (rather than constructing sampling points) of the set Z in the case of p(Y)=sum_i Y_i^2 and homogeneous Q. A by-product of our procedure is a bound (dn)^O(k) on the number of connected components of Z. The procedure consists of exact symbolic computations in D and outputs vectors of algebraic numbers. It involves extending K by infinitesimals and subsequent limit computation by a novel procedure that utilizes knowledge of an explicit isomorphism between real algebraic sets.