Bezout theorem for Nash functions
type: Item
instance of: scholarly article
title: Bezout theorem for Nash functions
zbMATH Open document ID: 0704.14045
DOI: 10.1016/0022-4049(89)90080-7
author: R. Ramanakoraisina
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \(V=\{x\in {\mathbb{R}}^ n:\) \(f_ 1(x)=...=f_ p(x)=0\}\). Milnor and Thom gave an upper bound for the sum of the Betti numbers of V in the case where V is compact and \(f_ i\) denotes a polynomial map such that for each \(i=1,...,p\), \(\deg (f_ i)\leq d:\sum^{n}_{i=0}b_ i(V) \leq d(2d-1)^{n-1}.\) Our aim is to obtain a similar relation in a larger class of functions than the polynomials: Nash functions \((C^{\infty}\quad semi-algebraic\) functions, defined in an open semi-algebraic U of \(R^ n\), where R denotes a real closed field). It is a well-known result that Nash functions verify a polynomial equation: \(P(x,f(x))=0\) where \(P\in R[X,Y]\) does not vanish identically. So we define the complexity of a Nash function as being the total degree of its minimal polynomial. This notion will replace the degree of polynomial in Milnor-Thom's formula. We give a Bezout theorem which uses this complexity. Then, we obtain an upper bound for the sum of the Betti numbers of a compact Nash set V:\(\sum^{n}_{i=0}b_ i(V) \leq d^{2n-1}\) where \(f_ i\) is a Nash function of complexity less than or equal to d. In the case where V is not compact, we have: \(\sum^{n}_{i=0}b_ i(V) \leq (2^{p+1}d^ p)^{2n-1}\).
Mathematics Subject Classification ID: 14P20
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 4154637
zbMATH Keywords: complexity of a Nash function
zbMATH Keywords: Bezout theorem
zbMATH Keywords: sum of the Betti numbers
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90080-7
OpenAlex ID: W2095309112
cites work: Q4204256
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description: scientific article; zbMATH DE number 4154637