On the geometry of polar varieties

Abstract

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces.In particular, we show the non-emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment.

Point searching in real singularcomplete intersection varieties: algorithms of intrinsic complexity

“…The procedure Π i is based on a geometrical and computational analysis of the dual polar varieties of the two families of incidence varieties (see [3,4,5] for the notion of a dual polar variety). These geometric objects are called bipolar varieties of S .…”

“…. , F p ) -regular points (see [5], Corollary 2 and Section 3.1). This motivates the consideration of the so-called generic polar varieties W K(a) (S) and W K(a) (S) , associated with complex ((n − p − i + 1) × (n + 1)) -matrices a which are generic in the above sense, as invariants of the complex variety S (independently of the given equation system F 1 = 0, .…”

“…, F p = 0 ). However, when a generic ((n − p − i + 1) × (n + 1)) -matrix a is real, we cannot consider W K(a) (S R ) and W K(a) (S R ) as invariants of the real variety S R , since for suitable real generic ((n − p − i + 1) × (n + 1)) -matrices these polar varieties may turn out to be empty, whereas for other real generic matrices they may contain points (see [5], Theorem 1 and Corollary 2 and [8], Theorem 8 and Corollary 9).…”

“…About 10 years ago they became also a fundamental tool for the design of pseudo-polynomial computer procedures with intrinsic complexity bounds which find for a given complete intersection variety S with a smooth real trace S R algebraic sample points for each connected component of S R . For details we refer to [41] and [5]. We fix for the rest of the paper a strongly reduced regular sequence the Jacobian of F .…”

Polar, bipolar and copolar varieties: Real solving of algebraic varieties with intrinsic complexity

“…, F p = 0). However, when a generic ((n − p − i + 1) × (n + 1))-matrix a is real, we cannot consider W K(a) (S R ) and W K(a) (S R ) as invariants of the real variety S R , since for suitable real generic ((n − p − i + 1) × (n + 1))-matrices these polar varieties may turn out to be empty, whereas for other real generic matrices they may contain points (see [5], Theorem 1 and Corollary 2 and [6], Theorem 8 and Corollary 9). In case that S R is smooth and a is a real ((n − p − i + 1) × (n + 1))-matrix, the real dual polar variety W K(a) (S R ) contains at least one point of each connected component of S R , whereas the classic (complex or real) polar varieties W K(a) (S) and W K(a) (S R ) may be empty (see [3] and [4], Proposition 2).…”