18,794 research outputs found
Computing Tropical Varieties
The tropical variety of a -dimensional prime ideal in a polynomial ring
with complex coefficients is a pure -dimensional polyhedral fan. This fan is
shown to be connected in codimension one. We present algorithmic tools for
computing the tropical variety, and we discuss our implementation of these
tools in the Gr\"obner fan software \texttt{Gfan}. Every ideal is shown to have
a finite tropical basis, and a sharp lower bound is given for the size of a
tropical basis for an ideal of linear forms.Comment: 22 pages, 2 figure
Irreducibility criterion for algebroid curves
The purpose of this paper is to give an algorithm for deciding the
irreducibility of reduced algebroid curves over an algebraically closed field
of arbitrary characteristic. To do this, we introduce a new notion of local
tropical variety which is a straightforward extension of tropism introduced by
Maurer, and then give irreducibility criterion for algebroid curves in terms
local tropical varieties. We also give an algorithm for computing the
value-semigroups of irreducible algebroid curves. Combining the irreducibility
criterion and the algorithm for computing the value-semigroups, we obtain an
algorithm for deciding the irreducibility of algebroid curves.Comment: 20 pages, v3: major revisio
A-Tint: A polymake extension for algorithmic tropical intersection theory
In this paper we study algorithmic aspects of tropical intersection theory.
We analyse how divisors and intersection products on tropical cycles can
actually be computed using polyhedral geometry. The main focus of this paper is
the study of moduli spaces, where the underlying combinatorics of the varieties
involved allow a much more efficient way of computing certain tropical cycles.
The algorithms discussed here have been implemented in an extension for
polymake, a software for polyhedral computations.Comment: 32 pages, 5 figures, 4 tables. Second version: Revised version, to be
published in European Journal of Combinatoric
A Tropical F5 algorithm
Let K be a field equipped with a valuation. Tropical varieties over K can be
defined with a theory of Gr{\"o}bner bases taking into account the valuation of
K. While generalizing the classical theory of Gr{\"o}bner bases, it is not
clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to
the tropical case. Among them, one of the most efficient is the celebrated F5
Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous
ideals, it can be adapted to the tropical case. We prove termination and
correctness. Because of the use of the valuation, the theory of tropical
Gr{\"o}b-ner bases is promising for stable computations over polynomial rings
over a p-adic field. We provide numerical examples to illustrate
time-complexity and p-adic stability of this tropical F5 algorithm
Algorithms for Mumford curves
Mumford showed that Schottky subgroups of give rise to certain
curves, now called Mumford curves, over a non-Archimedean field K. Such curves
are foundational to subjects dealing with non-Archimedean varieties, including
Berkovich theory and tropical geometry. We develop and implement numerical
algorithms for Mumford curves over the field of p-adic numbers. A crucial and
difficult step is finding a good set of generators for a Schottky group, a
problem solved in this paper. This result allows us to design and implement
algorithms for tasks such as: approximating the period matrices of the
Jacobians of Mumford curves; computing the Berkovich skeleta of their
analytifications; and approximating points in canonical embeddings. We also
discuss specific methods and future work for hyperelliptic Mumford curves.Comment: 32 pages, 4 figure
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