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Curve Valuations and Mixed Volumes in the Implicitization of Rational Varieties

Alicia Dickenstein 1 Maria Isabel Herrero 1 Bernard Mourrain 2
2 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA | UoA - National and Kapodistrian University of Athens = University of Athens
Abstract : We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.
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https://hal.inria.fr/hal-02958248
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Submitted on : Monday, October 5, 2020 - 4:55:28 PM
Last modification on : Tuesday, October 6, 2020 - 3:29:08 AM

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Alicia Dickenstein, Maria Isabel Herrero, Bernard Mourrain. Curve Valuations and Mixed Volumes in the Implicitization of Rational Varieties. 2020. ⟨hal-02958248⟩

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