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

• Main Authors: Grigoriev, Dima., Pasechnik, Dmitrii V.
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2011
• Subjects: DRNTU::Science::Mathematics::Applied mathematics
• Online Access: https://hdl.handle.net/10356/92360; http://hdl.handle.net/10220/6869
• Institution: Nanyang Technological University

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 com¬putes, 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 constructing sampling points) of the set Z in the case of p(Y ) = Σi Y i2 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.