Local extrema of analytic functions

Abstract

We give a complete answer to the problem of the finite decidability of the local extremality character of a real analytic function at a given point, a problem that found partial answers in some works by Severi and Łojasiewicz. Consider a real analytic functionf defined in a neighbourhood of a pointx ₀∈R ⁿ. Restrictf to the spherical surface centered inx ₀ and with radiusr≥0 and take its infimumm(r) and its supremumM(r). We establish some properties ofm(r) andM(r) for smallr>0. In particular, we prove that they have asymptotic expansions of the formf(x ₀)+c·(r ^α+o(r ^α)) asr→0 for a realc and a rational α≥1 (of course the parameters will usually be different form andM).