Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values

Abstract

Let X ⊂ C n be an affine variety and f : X → C m be the restriction to X of a polynomial map C n → C m. We construct an affine Whitney stratification of X. The set K (f) of stratified generalized critical values of f can also be computed. We show that K (f) is a nowhere dense subset of C m which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.

Conormal Spaces and Whitney Stratifications

“…Related Work. The current state of the art in this area appears to be the recent work of Ðinh and Jelonek [23]. To the best of our knowledge, given a complex algebraic variety X embedded in n-dimensional affine or projective space, all prior stratification methods (such as [28,29] for instance) require quantifier elimination in approximately 4n real variables.…”

“…The work of [23] improves upon such cylindrical algebraic decomposition (CAD) approaches by requiring only Gröbner basis (GB) computations in approximately 4n complex variables. The advantage enjoyed by GB methods over their CAD counterparts in practice has been well-documented [38,10].…”

“…Our algorithm further reduces the Whitney stratification problem to Gröbner basis type computations in approximately 2n complex variables, and additionally, is able to preserve the sparsity structure of the input in ways that make a significant difference to real-world performance. By contrast, the algorithm of [23] requires various choices of generic linear forms, which end up removing a lot of sparsity from the systems being considered. As an added bonus our algorithm is completely deterministic, while the algorithm of [23] is probabilistic.…”

“…By contrast, the algorithm of [23] requires various choices of generic linear forms, which end up removing a lot of sparsity from the systems being considered. As an added bonus our algorithm is completely deterministic, while the algorithm of [23] is probabilistic. We have implemented both algorithms, and provide a performance comparison in the final Section of this paper.…”

“…Our implementation is in Macaulay2 [19], and can be found along with documentation at the link below: http://martin-helmer.com/Software/WhitStrat As remarked in the Introduction, the state of the art for Whitney stratification algorithms appears to be the recent algorithm of [23, §2]. We are not aware of any existing implementations of earlier algorithms based on cylindrical algebraic decomposition, and in any event we would expect their performance to be slower than that of [23] overall. Since the authors of [23] did not provide an implementation of their algorithm, we have implemented it ourselves in Macaulay2.…”