Real tropicalization and negative faces of the Newton polytope

Abstract

In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f1<0,…,fk<0} defined in the positive orthant and the combinatorial properties of the defining polynomials f1,…,fk. We describe a cone that depends only on the face structure of the Newton polytopes of f1,…,fk and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if S={f<0} and the polynomial f has generic coefficients. Furthermore, we show that for a maximally sparse polynomial f the real tropicalization of S={f<0} is determined by the outer normal cones of the Newton polytope of f and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant.