Real tropicalization and negative faces of the Newton polytope

In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f₁<0,…,f_k<0} defined in the positive orthant and the combinatorial properties of the defining polynomials f₁,…,f_k. We describe a cone that depends only on the face structure of the Newton polytopes of f₁,…,f_k 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.