Abstract We generalize the seminal polynomial partitioning theorems of Guth and Katz 33, 28 to a set of semi-Pfaffian sets. Specifically, given a set Rⁿ of k -dimensional semi-Pfaffian sets, where each is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain q of length r, for any D 1, we prove the existence of a polynomial P RX₁, , Xₙ of degree at most D such that each connected component of Rⁿ Z (P) intersects at most | |D^{n - k - r} elements of. Also, under some mild conditions on q, for any D 1, we prove the existence of a Pfaffian function P' of degree at most D defined with respect to q, such that each connected component of Rⁿ Z (P') intersects at most | |D^{n-k} elements of. To do so, given a k -dimensional semi-Pfaffian set X Rⁿ, and a polynomial P RX₁, , Xₙ of degree at most D, we establish a uniform bound on the number of connected components of Rⁿ Z (P) that X intersects; that is, we prove that the number of connected components of (Rⁿ Z (P) ) X is at most <jats: inline-graphic xmlns: xlink="ht
Lotz et al. (Thu,) studied this question.