Subtleties of the minmax selector

TL;DR

This paper investigates the conditions under which minmax and maxmin critical values of quadratic functions coincide, highlighting the dependence on the choice of coefficients and providing counterexamples.

Contribution

It establishes the equality of minmax and maxmin critical values over fields and demonstrates their dependence on the coefficient ring with a counterexample.

Findings

Minmax and maxmin critical values are equal over fields for quadratic functions.

Counterexample shows dependence on the coefficient ring.

Equality may fail when coefficients are in a ring, not a field.

Abstract

In this note, we show that the minmax and maxmin critical values of a function quadratic nondegenerate at infinity are equal when defined in homology or cohomology with coefficients in a field. However, by an example of F.Laudenbach, this is not always true for coefficients in a ring and, even in the case of a field, the minmax-maxmin depends on the field.