Bounding the Number of Stable Homotopy Types of a Parametrized Family of Semi-algebraic Sets Defined by Quadratic Inequalities
نویسندگان
چکیده
We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in R, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in `. More precisely, we prove the following. Let R be a real closed field and let P = {P1, . . . , Pm} ⊂ R[Y1, . . . , Y`,X1, . . . , Xk], with degY (Pi) ≤ 2,degX(Pi) ≤ d, 1 ≤ i ≤ m. Let S ⊂ R `+k be a semialgebraic set, defined by a Boolean formula without negations, whose atoms are of the form, P ≥ 0, P ≤ 0, P ∈ P . Let π : R → R be the projection on the last k co-ordinates. Then the number of stable homotopy types amongst the fibers Sx = π−1(x) ∩ S is bounded by (2 `kd).
منابع مشابه
Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial
In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We ...
متن کاملOn Homotopy Types of Limits of Semi-algebraic Sets and Additive Complexity of Polynomials
We prove that the number of homotopy types of limits of oneparameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifierfree first order formula defining the family. As an important consequence, we derive that the number of homotopy types of semi-algebraic subsets of R defined by a quantifier-free first o...
متن کاملA Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities
In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semi-algebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0, . . . , Ps ≥ 0, where Pi ∈ R[X1, . . . , Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 2 + (k − s) + 1 2 · min{s+1,k−i}
متن کاملA new family in the stable homotopy groups of spheres
Let $p$ be a prime number greater than three. In this paper, we prove the existence of a new family of homotopy elements in the stable homotopy groups of spheres $pi_{ast}(S)$ which is represented by $h_nh_mtilde{beta}_{s+2}in {rm Ext}_A^{s+4, q[p^n+p^m+(s+2)p+(s+1)]+s}(mathbb{Z}_p,mathbb{Z}_p)$ up to nonzero scalar in the Adams spectral sequence, where $ngeq m+2>5$, $0leq sExt}_A^{s+2,q[(s+2)p...
متن کاملOn the number of homotopy types of fibres of a definable map
In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map in terms of the format of its graph. In particular, we show that if a semi-algebraic set S ⊂ Rm+n, where R is a real closed field, is defined by a Boolean formula with s polynomials of degree less than d, and π : Rm+n → Rn is the projection on a subspace, then the num...
متن کامل