Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities
Saugata Basu, Michael Kettner · Proceedings of the London Mathematical Society · 2008
We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in Rℓ, 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 = {P1, … , Pm} ⊂ R[Y1, … ,Yℓ,X1, … ,Xk], with degY(Pi) ⩽ 2, degX(Pi) ⩽ d, 1 ⩽ i ⩽ m. Let S ⊂ Rℓ+k be a semi-algebraic set, defined by a Boolean formula without negations, with atoms of the form P ⩾ 0, P ⩽ 0, P ∈ 𝒫. Let π : Rℓ+k → Rk be the projection on the last k coordinates. Then the number of stable homotopy types amongst the fibers Sx = π−1(x) ∩ S is bounded by (2mℓkd)O(mk).