2018/03/15 by Fernando, José F.
#03C64 (secondary) #14P15 (primary) #26E05 #32B20 #54C30 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1803.06017
Let \mathfrak F be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical and basic topological operations. Let M be a real analytic manifold and denote \mathfrak F(M) the family of the subsets of M that belong to \mathfrak F. Let f:X→\mathbb R be a subanalytic function on a subset X∈\mathfrak F(M) such that the inverse image under f of each interval of \mathbb R belongs to \mathfrak F(M). Let \rm Max(f) be the set of local maxima of f and consider \rm Maxλ(f):=\rm Max(f)∩\f=λ\ for each λ∈\mathbb R. If f is continuous, then \rm Max(f)=\bigsqcup_λ∈\mathbb R\rm Maxλ(f)∈\mathfrak F(M) if and only if the family \\rm Maxλ(f)\_λ∈\mathbb R is locally finite in M. If we erase continuity condition, there exist subanalytic functions f:X→ M such that \rm Max(f)∈\mathfrak F(M), but the family \\rm Maxλ(f)\_λ∈\mathbb R is not locally finite in M or such that \rm Max(f) is connected but it is not even subanalytic. If \mathfrak F is the category of subanalytic sets and f:X→\mathbb R is a subanalytic map f that maps relatively compact subsets of M contained in X to bounded subsets of \mathbb R, then \rm Max(f)∈\mathfrak F(M) and the family \\rm Maxλ(f)\_λ∈\mathbb R is locally finite in M. If the category \mathfrak F contains the intersections of algebraic sets with real analytic submanifolds and X∈\mathfrak F(M) is not closed in M, there exists a continuous subanalytic function f:X→\mathbb R with graph belonging to \mathfrak F(M×\mathbb R) such that inverse images under f of the intervals of \mathbb R belong to \mathfrak F(M) but \rm Max(f) does not belong to \mathfrak F(M).