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- W2125297716 abstract "Let $R$ be a real closed field, $ {mathcal Q} subset R[Y_1,...,Y_ell,X_1,...,X_k], $ with $ deg_{Y}(Q) leq 2, deg_{X}(Q) leq d, Q in {mathcal Q}, #({mathcal Q})=m,$ and $ {mathcal P} subset R[X_1,...,X_k] $ with $deg_{X}(P) leq d, P in {mathcal P}, #({mathcal P})=s$, and $S subset R^{ell+k}$ a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P geq 0, P leq 0, P in {mathcal P} cup {mathcal Q}$. We prove that the sum of the Betti numbers of $S$ is bounded by [ ell^2 (O(s+ell+m)ell d)^{k+2m}. ] This is a common generalization of previous results on bounding the Betti numbers of closed semi-algebraic sets defined by polynomials of degree $d$ and 2, respectively. We also describe an algorithm for computing the Euler-Poincar'e characteristic of such sets, generalizing similar algorithms known before. The complexity of the algorithm is bounded by $(ell s m d)^{O(m(m+k))}$." @default.
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- W2125297716 date "2009-01-01" @default.
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- W2125297716 title "On Semidefinite Programming Relaxations of the Traveling Salesman Problem" @default.
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- W2125297716 doi "https://doi.org/10.1137/070711141" @default.
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