2,488 research outputs found
Residues and tame symbols on toroidal varieties
We introduce a new approach to the study of a system of algebraic equations
in the algebraic torus whose Newton polytopes have sufficiently general
relative positions. Our method is based on the theory of Parshin's residues and
tame symbols on toroidal varieties. It provides a uniform algebraic explanation
of the recent result of Khovanskii on the product of the roots of such systems
and the Gel'fond--Khovanskii result on the sum of the values of a Laurent
polynomial over the roots of such systems, and extends them to the case of an
algebraically closed field of arbitrary characteristic.Comment: 26 pages, minor changes, title changed, new introduction, references
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Specht Polytopes and Specht Matroids
The generators of the classical Specht module satisfy intricate relations. We
introduce the Specht matroid, which keeps track of these relations, and the
Specht polytope, which also keeps track of convexity relations. We establish
basic facts about the Specht polytope, for example, that the symmetric group
acts transitively on its vertices and irreducibly on its ambient real vector
space. A similar construction builds a matroid and polytope for a tensor
product of Specht modules, giving "Kronecker matroids" and "Kronecker
polytopes" instead of the usual Kronecker coefficients. We dub this process of
upgrading numbers to matroids and polytopes "matroidification," giving two more
examples. In the course of describing these objects, we also give an elementary
account of the construction of Specht modules different from the standard one.
Finally, we provide code to compute with Specht matroids and their Chow rings.Comment: 32 pages, 5 figure
Convergent Puiseux Series and Tropical Geometry of Higher Rank
We propose to study the tropical geometry specifically arising from
convergent Puiseux series in multiple indeterminates. One application is a new
view on stable intersections of tropical hypersurfaces. Another one is the
study of families of ordinary convex polytopes depending on more than one
parameter through tropical geometry. This includes cubes constructed by
Goldfarb and Sit (1979) as special cases.Comment: 32 pages, 3 figure
Zeros of random tropical polynomials, random polytopes and stick-breaking
For , let be independent and identically
distributed random variables with distribution with support .
The number of zeros of the random tropical polynomials is also the number of faces of the lower convex
hull of the random points in . We show that this
number, , satisfies a central limit theorem when has polynomial decay
near . Specifically, if near behaves like a
distribution for some , then has the same asymptotics as the
number of renewals on the interval of a renewal process with
inter-arrival distribution . Our proof draws on connections
between random partitions, renewal theory and random polytopes. In particular,
we obtain generalizations and simple proofs of the central limit theorem for
the number of vertices of the convex hull of uniform random points in a
square. Our work leads to many open problems in stochastic tropical geometry,
the study of functionals and intersections of random tropical varieties.Comment: 22 pages, 5 figure
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