2023/06/08 by Máté L. Telek, Telek, Máté L.
Computer Science · #Algebraic Geometry (math.AG) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2306.05154
openalex publication_date 2023/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.