Refined bounds on the number of connected components of sign conditions on a variety

Let R be a real closed field, P, Q ⊂ R[X1, . . . , Xk] finite subsets of polynomials, with the degrees of the polynomials in P (resp. Q) bounded by d (resp. d0). Let V ⊂ Rk be the real algebraic variety defined by the polynomials in Q and suppose that the real dimension of V is bounded by k . We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family P on V is bounded by kX j=0 “8s j ” (2k+1 + 1)(2d0)k−k +1 dj max{2d0, d}k −j . In case 2d0 ≤ d, the above bound can be written simply as kX j=0 “8s j ” dk−k +1 0 dk O(1)k . This improves in certain cases (when d0 d) the previous best known bound of X 1≤j≤k “s j ” 4j d(2d − 1)k−1 , on the same number proved in [4] in the case d = d0.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Sal Barone, Saugata Basu
• Keywords: CORR 2011, Education, Polynomials, Real Algebraic Variety, Realizable Sign Conditions