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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  1 , … ,  P   m  } ⊂  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  ⊂  R  ℓ+ k   be a semi‐algebraic set, defined by a Boolean formula without negations, with atoms of the form  P  ⩾ 0,  P  ⩽ 0,  P  ∈ . Let π :  R  ℓ+ k   →  R   k   be the projection on the last  k  coordinates. Then the number of stable homotopy types amongst the fibers  S   x   = π −1 ( x ) ∩  S  is bounded by (2  m  ℓ kd ) O( mk ) .

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