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Proceedings of the London Mathematical Society Advance Access originally published online on August 11, 2008
Proceedings of the London Mathematical Society 2009 98(2):298-324; doi:10.1112/plms/pdn031
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© 2008 London Mathematical Society

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

Saugata Basu and Michael Kettner

School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332
USA
mkettner@math.gatech.edu

Received 24 August 2007.

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in R{ell}, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in {ell}. More precisely, we prove the following. Let R be a real closed field and let P = {P1, ... , Pm} sub R[Y1, ... ,Y{ell},X1, ... ,Xk], with degY(Pi) ≤ 2, degX(Pi) ≤ d, 1 ≤ i ≤ m. Let S sub R{ell}+k be a semi-algebraic set, defined by a Boolean formula without negations, with atoms of the form P ≥ 0, P ≤ 0, P isin P. Let {pi} : R{ell}+k -> Rk be the projection on the last k coordinates. Then the number of stable homotopy types amongst the fibers Sx = {pi}–1(x) {cap} S is bounded by (2m{ell}kd)O(mk).

2000 Mathematics Subject Classification 14P10, 14P25 (primary), 55P15, 55P42 (secondary).

The first author was supported in part by an NSF grant CCF-0634907. Part of this work was done when the authors were visiting the Institute of Mathematics and Its Application, Minneapolis.


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