Coercive Polynomials and Their Newton Polytopes
Tomáš Bajbar, Oliver T. Stein · SIAM Journal on Optimization · 2015
Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this article, we analyze the coercivity on $\mathbb{R}^n$ of multivariate polynomials $f\in \mathbb{R}[x]$ in terms of their so-called Newton polytopes at infinity. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions solely containing information about the geometry of the vertex set of the Newton polytope at infinity, as well as sign conditions on the corresponding polynomial coefficients. For all other polynomials, the so-called gem irregular polynomials, we introduce sufficient conditions for coercivity based on those from the regular case. For some special cases of gem irregular polynomials, we establish necessary conditions for coercivity, too. Using our techniques, the problem of deciding the coercivity of a polynomial can often be studied based on its Newton polytope at infinity. We relate our results to the context of polynomial optimization theory and the existing literature therein, and we illustrate our results with several examples.