POLYNOMIAL-TIME COMPUTING OVER QUADRATIC MAPS I. SAMPLING IN REAL ALGEBRAIC SETS
Dima Grigoriev, Dmitrii V. Pasechnik · 2005
Abstract. Given a quadratic map Q: Kn → Kk defined over a computable subring D of a real closed field K, and p ∈ D[Y1,...,Yk] of degree d we consider the zero set Z = Z(p(Q(X)), Kn) ⊆ Kn of p(Q(X1,...,Xn)) ∈ D[X1,...,Xn]. 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 Kn 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 is only capable of deciding in nO(k2) operations the nonemptiness (rather than i Y 2 i constructing sampling points) of the set Z in the case of p(Y) = ∑ 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.