منابع مشابه
Betti Numbers of Semialgebraic and Sub-pfaffian Sets
Let X be a subset in [−1, 1]n0 ⊂ Rn0 defined by a formula X = {x0|Q1x1Q2x2 . . . Qνxν((x0,x1, . . . ,xν) ∈ Xν)}, where Qi ∈ {∃,∀}, Qi 6= Qi+1, xi ∈ Rni , and Xν be either an open or a closed set in [−1, 1]n0+...+nν being a difference between a finite CW -complex and its subcomplex. We express an upper bound on each Betti number of X via a sum of Betti numbers of some sets defined by quantifier-...
متن کاملBounding the equivariant Betti numbers of symmetric semi-algebraic sets
Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of R in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Olĕınik and Petrovskĭı, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several ap...
متن کاملComplexity of Stratifications of Semi-Pfaffian Sets
An effective algorithm for a smooth (weak) stratification of a real semi-Pfaffian set is suggested, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. An explicit estimate of complexity of the algorithm and of the resulting stratification is given, in terms of the parameters of the Pfaffian functions defining the original semiPfaffian set. The a...
متن کاملComputing the Betti Numbers of Semi-algebraic Sets Defined by Partly Quadratic Systems of Polynomials
Let R be a real closed field, Q ⊂ R[Y1, . . . , Y`, X1, . . . , Xk], with degY (Q) ≤ 2, degX(Q) ≤ d,Q ∈ Q,#(Q) = m, and P ⊂ R[X1, . . . , Xk] with degX(P ) ≤ d, P ∈ P,#(P) = s. Let S ⊂ R`+k be a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We describe an algorithm for computing the the Betti numbers of S generalizing a similar alg...
متن کاملBetti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials
Let R be a real closed field, Q ⊂ R[Y1, . . . , Y`, X1, . . . , Xk], with degY (Q) ≤ 2, degX(Q) ≤ d,Q ∈ Q,#(Q) = m, and P ⊂ R[X1, . . . , Xk] with degX(P ) ≤ d, P ∈ P,#(P) = s, and S ⊂ R`+k a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We prove that the sum of the Betti numbers of S is bounded by (`smd)O(m+k). This is a common ge...
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ژورنال
عنوان ژورنال: Journal of Pure and Applied Algebra
سال: 1999
ISSN: 0022-4049
DOI: 10.1016/s0022-4049(99)00017-1