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Register a new userCurve valuations and mixed volumes in the implicitization of rational varieties
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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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The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When the image is a hypersurface, the output is the Newton polytope of the defining polynomial. TrIm can thus be used to compute mixed fiber polytopes, including secondary polytopes.
The integral model of a GU(n-1,1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the Gillet-Soule arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L-functions.
We give Macaulay2 algorithms for computing mixed multiplicities of ideals in a polynomial ring. This enables us to find mixed volumes of lattice polytopes and sectional Milnor numbers of a hypersurface with an isolated singularity. The algorithms use the defining equations of the multi-Rees algebra of ideals. We achieve this by generalizing a recent result of David A. Cox, Kuei-Nuan Lin, and Gabriel Sosa in. One can also use a Macaulay2 command `reesIdeal to calculate the defining equations of the Rees algebra. We compare the computation time of our scripts with the scripts already available.
The paper has two goals: the study the associated graded ring of contracted homogeneous ideals in $K[x,y]$ and the study of the Groebner fan of the ideal $P$ of the rational normal curve in ${bf P}^d$. These two problems are, quite surprisingly, very tightly related. We completely classify the contracted ideals with a Cohen-Macaulay associated graded rings in terms of the numerical invariants arising from Zariskis factorization. We determine explicitly all the initial ideals (monomial or not) of $P$ that are Cohen-Macaulay.
We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, ElimTH, has as main step the computation of an elimination ideal via a textit{truncated, homogeneous} Grobner basis. The other algorithm, Direct, computes the implicitization directly using an approach inspired by the generalized Buchberger-Moller algorithm. Either may be used inside the third algorithm, RatPar, to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm ModImplicit, for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.
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