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Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities.
- Source :
- Proceedings of the London Mathematical Society; Mar2009, Vol. 98 Issue 2, p298-298, 1p
- Publication Year :
- 2009
-
Abstract
- We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in Rℓ, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in ℓ. More precisely, we prove the following. Let R be a real closed field and let ℘ = {P1,ââ¦â, Pm} â R[Y1,ââ¦â,Yℓ,X1,ââ¦â,Xk], with degY(Pi) ⤠2, degX(Pi) ⤠d, 1 ⤠i ⤠m. Let S â Rℓ be a semi-algebraic set, defined by a Boolean formula without negations, with atoms of the form P ⥠0, P ⤠0, P â ℘. Let Ï : Rℓ â Rk be the projection on the last k coordinates. Then the number of stable homotopy types amongst the fibers Sx = Ïâ1(x) â© S is bounded by (2mℓkd)O(mk). [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00246115
- Volume :
- 98
- Issue :
- 2
- Database :
- Complementary Index
- Journal :
- Proceedings of the London Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 36783731
- Full Text :
- https://doi.org/10.1112/plms/pdn031