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

Abstract.

Given a quadratic map $Q:\mathbb{K}^n \to \mathbb{K}^k$ defined over a computable subring D of a real closed field $\mathbb{K},$ and p ∈D[Y₁,...,Y _k ] of degree d, we consider the zero set $Z = Z(p(Q(X)), \mathbb{K}^n) \subseteq \mathbb{K}^n$ of p(Q(X₁,...,X _n )) ∈D[X₁,...,X _n ]. We present a procedure that computes, in (dn)^O(k) arithmetic operations in D, a set $\mathcal{S}$ of (real univariate representations of) sampling points in $\mathbb{K}^n$ that intersects nontrivially each connected component of Z. As soon as k=o(n), this is faster than the known 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 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)=∑ _i Y ² _i 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 $\mathbb{K}$ by infinitesimals and subsequent limit computation by a novel procedure that utilizes knowledge of an explicit isomorphism between real algebraic sets.