Tropical Implicitization and Mixed Fiber Polytopes

1. Introduction. Implicitization is the problem of transforming a given parametric representation of an algebraic variety into its implicit representation as the zero set of polynomials. Most algorithms for elimination and implicitization are based on multivariate resultants or Gröbner bases, but current implementations of these methods are often too slow. When the variety is a hypersurface, the coefficients of the implicit equation can also be computed by way of numerical linear algebra [3,7], provided the Newton polytope of that implicit equation can be predicted a priori.

The problem of predicting the Newton polytope was recently solved independently by three sets of authors, namely, by Emiris, Konaxis and Palios [8], Esterov and Khovanskii [13], and in our joint papers with Tevelev [18,19]. A main conclusion of these papers can be summarized as follows:

The Newton polytope of the implicit equation is a mixed fiber polytope.

The first objective of the present article is to explain this conclusion and to present the software package TrIm for computing such mixed fiber polytopes. The name of our program stands for Tropical Implicitization, and it underlines our view that the prediction of Newton polytopes is best understood within the larger context of tropical algebraic geometry. The general theory of tropical elimination developed in [18] unifies earlier results on discriminants [4] and on generic polynomial maps whose images can have any codimension [19]. The second objective of this article is to explain the main results of tropical elimination theory and their implementation in TrIm. Numerous hands-on examples will illustrate the use of the software. At various places we give precise pointers to [8] and [13], so as to highlight similarities and differences among the different approaches to the subject.

Our presentation is organized as follows. In Section 2 we start out with a quick guide to TrIm by showing some simple computations. In Section 3 we explain mixed fiber polytopes. That exposition is self-contained and may be of independent interest to combinatorialists. In Section 4 we discuss the computation of mixed fiber polytopes in the context of elimination theory, and in Section 5 we show how the tropical implicitization problem is solved in TrIm. Theorem 5.1 expresses the Newton polytope of the implicit equation as a mixed fiber polytope. In Section 6 we present results in tropical geometry on which the development of TrIm is based, and we explain various details concerning our algorithms and their implementation.

2. How to use TrIm. The first step is to download TrIm from the following website which contains information for installation in Linux: http://math.mit.edu/ ~jyu/TrIm TrIm is a collection of C++ programs which are glued together and integrated with the external software polymake [9] using perl scripts. The language perl was chosen for ease of interfacing between various programs.

The fundamental problem in tropical implicitization is to compute the Newton polytope of a hypersurface which is parametrized by Laurent polynomials with sufficiently generic coefficients. As an example we consider the following three Laurent polynomials in two unknowns x and y with sufficiently generic coefficients α 1 , α 2 , α 3 , β 1 , β 2 , β 3 , γ 1 , γ 2 , γ 3 :

We seek the unique (up to scaling) irreducible polynomial F (u, v, w) which vanishes on the image of the corresponding morphism (C * ) 2 → C 3 . Using our software TrIm, the Newton polytope of the polynomial F (u, v, w) can be computed as follows. We first create a file input with the contents

Here the coefficients are suppressed: they are tacitly assumed to be generic. We next run a perl script using the command ./TrIm.prl input. The output produced by this program call is quite long. It includes the lines Ignoring the initial 1, this list consists of 13 lattice points in R 3 , and these are precisely the vertices of the Newton polytope of F (u, v, w). The above output format is compatible with the polyhedral software Polymake [9]. We find that the Newton polytope has 10 facets, 21 edges, and 13 vertices.

Further down in the output, TrIm prints a list of all lattice points in the Newton polytope, and it ends by telling us the number of lattice points:

Each of the 383 lattice points (i, j, k) represents a monomial u i v j w k which might occur with non-zero coefficient in the expansion of F (u, v, w). Hence, to recover the coefficients of F (u, v, w) we must solve a linear system of 382 equations with 383 unknowns. Interestingly, in this example, 39 of the 383 monomials always have coefficient zero in F (u, v, w). Even when α 1 , . . . , γ 3 are completely generic, the number of monomials in F (u, v, w) is only 344.

The command ./Trim.prl implements a certain algorithm, to be described in the next sections, whose input consists of n lattice polytopes in R n-1 and whose output consists of one lattice polytope in R n . In our example, wi

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