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

摘要

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-l, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in l. More precisely, we prove the following. Let R be a real closed field and let P = {P-1, ..., P-m} subset of R[Y-1, ..., Y-l, X-1, ..., X-k], with deg(Y) (P-i) <= 2, deg(X)(P-i) <= d, 1 <= i <= m. Let S subset of Rl+k be a semi-algebraic set, defined by a Boolean formula without negations, with atoms of the form P >= 0, P <= 0, P is an element of P. Let pi : Rl+k -> R-k be the projection on the last k coordinates. Then the number of stable homotopy types amongst the fibers S-x = pi(-1)(x) boolean AND S is bounded by (2(m) lkd)(O(mk)).