Exponential convexifying of polynomials

Abstract

Let X⊂Rn be a convex closed and semialgebraic set and let f be a polynomial positive on X. We prove that there exists an exponent N⩾1, such that for any ξ∈Rn the function φN(x)=eN|x−ξ|2f(x) is strongly convex on X. When X is unbounded we have to assume also that the leading form of f is positive in Rn∖{0}. We obtain strong convexity of ΦN(x)=eeN|x|2f(x) on possibly unbounded X, provided N is sufficiently large, assuming only that f is positive on X. We apply these results for searching critical points of polynomials on convex closed semialgebraic sets.