Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this article we analyze the coercivity on R n of multivariate polynomials f ∈ R[x] in terms of their Newton polytopes. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions imposed on the vertex set of their Newton polytopes. These conditions solely contain information about the geometry of the vertex set of the Newton polytope, 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 be reduced to the analysis of its Newton polytope. We relate our results to the context of the polynomial optimization theory and the existing literature therein, and we illustrate our results with several examples.
In this article we analyze the global diffeomorphism property of polynomial maps F : R n → R n by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials F 2 2 . This allows us to identify a class of polynomial maps F for which their global diffeomorphism property on R n is equivalent to their Jacobian determinant det JF vanishing nowhere on R n . In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.
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