The local-to-global minimum problem of Calvert and Vamanamurthy: a quartic counterexample in three variables and the cubic case in all dimensions

Abstract

In 1980 Calvert and Vamanamurthy asked whether a polynomial whose only critical point is a local minimum must attain its global minimum there. They gave counterexamples of every degree at least five in two or more variables and settled cubics in two and three variables and quartics in two variables, leaving open quartics in three or more variables and cubics in four or more variables. We resolve both remaining cases. First, we give an explicit quartic in three variables whose unique critical point is a nondegenerate strict local minimum, although the polynomial is unbounded below; the example extends to every dimension at least three, and its critical locus is a single point even over the complex numbers. Second, we prove that in every dimension a polynomial of exact degree three with a local minimum has another critical point, so that the cubic case holds vacuously; the proof is elementary and avoids the topological degree arguments used previously in two and three variables. Together with the results of Calvert and Vamanamurthy, this gives the complete classification of the local-to-global property by degree and dimension.