About the Euler-poincaré Characteristic of Semi-algebraic Sets Defined with Two Inequalities
نویسنده
چکیده
We express the Euler-Poincaré characteristic of a semi-algebraic set, which is the intersection of a non-singular complete intersection with two polynomial inequalities, in terms of the signatures of appropriate bilinear symmetric forms.
منابع مشابه
The Value Ring of Geometric Motivic Integration, and the Iwahori Hecke Algebra of Sl2 Ehud Hrushovski, David Kazhdan
In [1], an integration theory for valued fields was developed with a Grothendieck group approach. Two types of categories were studied. The first was of semi-algebraic sets over a valued field, with all semi-algebraic morphisms. The Grothendieck ring of this category was shown to admit two natural homomorphisms, esssentially into the Grothendieck ring of varieties over the residue field. These ...
متن کاملTopological Invariance of the Combinatorial Euler Characteristic of O-minimal Sets
We prove the topological invariance of the combinatorial Euler characteristic of ominimal sets with the help of a canonical, topologically defined stratification of o-minimal sets by locally compact ones. Introduction. Let Bn be the collection of semi-algebraic subsets of R, i.e. sets definable by a finite boolean combination of polynomial equalities and inequalities. The Bn, n ∈ N, satisfy (i)...
متن کاملComputing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time
For every fixed l > 0, we describe a singly exponential algorithm for computing the first l Betti number of a given semi-algebraic set. More precisely, we describe an algorithm that given a semi-algebraic set S ⊂ Rk a semi-algebraic set defined by a Boolean formula with atoms of the form P > 0, P < 0, P = 0 for P ∈ P ⊂ R[X1, . . . ,Xk], computes b0(S), . . . , bl(S). The complexity of the algor...
متن کاملEfficient algorithms for computing the Euler-Poincaré characteristic of symmetric semi-algebraic sets
We give algorithms with polynomially bounded complexities (for fixed degrees) for computing the generalized Euler-Poincaré characteristic of semi-algebraic sets defined by symmetric polynomials. This is in contrast to the best complexity of the known algorithms for the same problem in the non-symmetric situation, which is singly exponential. This singly exponential complexity for the latter pro...
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...
متن کامل