On the bifurcation sets of functions definable in o-minimal structures
Abstract
Let g:X -> Y be a smooth (i.e. C^\infty differentiable) map between two smooth manifolds. In analogy with the case of complex polynomial functions, we say that y_0 in Y is a typical value of g if there exists an open neighbourhood U of y_0 in Y, such that the restriction g:g^{-1}(U) -> U is a C^\infty trivial fibration. If y_0 in Y is not a typical value of g, then y_0 is called an atypical value of g. We denote by B_g the bifurcation set of g, i.e. the set of atypical values of g. In the case of a complex polynomial function f:C^n -> C it is known that B_f is a finite set. It was previously proved that the bifurcation sets of real polynomial functions are also finite. The aim of this note is to show that the bifurcation set B_f of a smooth definable function f:R^n -> R is finite .
Keywords
Cite
@article{arxiv.math/9706226,
title = {On the bifurcation sets of functions definable in o-minimal structures},
author = {Ta Lê Loi and Alexandru Zaharia},
journal= {arXiv preprint arXiv:math/9706226},
year = {2016}
}