On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S Rk+m be a compact semi-algebraic set defined by P1 0, . . . , P 0, where Pi R[X1, . . . , Xk, Y1, . . . , Ym], and deg(Pi) 2, 1 i . Let denote the standard projection from Rk+m onto Rm. We prove that for any q > 0, the sum of the first q Betti numbers of (S) is bounded by (k + m)O(q ) . We also present an algorithm for computing the the first q Betti numbers of (S), whose complexity is (k + m)2O(q ) . For fixed q and , both the bounds are polynomial in k + m.

• Type: Journal
• Year: 2008
• Where: DCG
• Authors: Saugata Basu, Thierry Zell
• Keywords: Algorithm, DCG 2008, Polynomial, Standard Projection