Curve valuations and mixed volumes in the implicitization of rational varieties

A. DICKENSTEIN, M. I. HERRERO, B. MOURRAIN

JOURNAL OF ALGEBRA

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2022 vol. 612 p. 691 - 721

0021-8693

We address the description of the tropicalization of families of rationalvarieties under parametrizations with prescribed support, via curve valuations.We recover and extend results by Sturmfels, Tevelev and Yu for genericcoefficients, considering rational parametrizations with non-trivialdenominator. The advantage of our point of view is that it can be generalizedto deal with non-generic parametrizations. We provide a detailed analysis ofthe degree of the closed image, based on combinatorial conditions on therelative positions of the supports of the polynomials defining theparametrization. We obtain a new formula and finer bounds on the degree, whenthe supports of the polynomials are different. We also present a new formulaand bounds for the order at the origin in case the closed image is ahypersurface.