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Tropical Effective Primary and Dual Nullstellens"atze
P. 379–391.
Grigoriev D., Podolskii V. V.
Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove tropical Nullstellensatz and moreover we show effective formulation of this theorem. Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and effective version is relevant for computational aspects of this field.
Language:
English
In book
Vol. 30. , Dagstuhl: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2015.
Быковский В. А., Романов М. А., Ustinov A., Чебышевский сборник 2021 Т. 22 № 1 С. 118–132
Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to ...
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Shkolnikov M., Kalinin N., European Journal of Mathematics 2019 Vol. 5 No. 3 P. 909–928
√√
Abstract Let f(a,b,c,d) = a2+b2 + c2+d2 − (a+c)2+(b+d)2, let
(a,b,c,d) stand for a,b,c,d ∈ Z0 such that ad − bc = 1. Defines
In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one.
We prove that F(s) converges when s > 1 and diverges at s = 1/2. We also prove that
1=1,(a,b,c,d) (a+c)2(b+d)2(a+b+c+d)2 3 ...
Added: August 15, 2019
Podolskii V. V., Grigoriev D., Discrete and Computational Geometry 2018 Vol. 59 No. 3 P. 507–552
Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows studying properties of mathematical objects such as algebraic varieties from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. ...
Added: October 16, 2018
Kalinin N., Guzmán-Sáenz A., Prieto Y. et al., Proceedings of the National Academy of Sciences of the United States of America 2018 Vol. 115 No. 35 P. E8135–E8142
Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models ...
Added: August 28, 2018
Podolskii V. V., Grigoriev D., Lecture Notes in Computer Science 2017 Vol. 10472 P. 284–297
Tropical algebra emerges in many fields of mathematics such as algebraic geometry, mathematical physics and combinatorial optimization. In part, its importance is related to the fact that it makes various parameters of mathematical objects computationally accessible. Tropical polynomials play an important role in this, especially for the case of algebraic geometry. On the other hand, ...
Added: December 19, 2017
Feigin E., Makhlin I., Fourier G. et al., / Series math "arxiv.org". 2017. No. 1711.00751.
We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type A. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for ...
Added: November 3, 2017
Kalinin N., Discrete and Computational Geometry 2017 Vol. 58 No. 1 P. 158–179
Suppose that there exists a hypersurface with the Newton polytope Δ, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular ...
Added: May 16, 2017
Shkolnikov M., Kalinin N., Comptes Rendus Mathematique 2016 Vol. 354 No. 2 P. 125–130
We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation ψ of the maximal stable state μ ≡ 3 is obtained by adding extra grains at several points. It appears that the result ψ◦ of the relaxation of ψ coincides with μ almost everywhere; the ...
Added: March 2, 2017
Kalinin N., Journal of Combinatorial Theory, Series A 2016 No. 137 P. 226–256
Let a planar algebraic curve C be defined over a valuation field by an equation F (x, y) = 0. Valuations of the coefficients of F define a subdivision of the Newton polygon Δ of the curve C. If a given point p is of multiplicity m on C, then the coefficients of F are subject to ...
Added: March 2, 2017
Shitov Y., Linear Algebra and its Applications 2016 Vol. 511 P. 19–21
Let K be a field with non-Archimedean valuation v, and assume A is a matrix of size m×n and rank k over K. Richter- Gebert, Sturmfels, and Theobald proved that the rows of A are a tropical basis of the corresponding linear space whenever m = k and any submatrix formed by k columns of ...
Added: September 7, 2016
Shitov Y., Linear Algebra and its Applications 2016 Vol. 499 P. 26–30
Let χ(A) denote the characteristic polynomial of a matrix A over a field; a standard result of linear algebra states that χ(A−1) is the reciprocal polynomial of χ(A). More formally, the condition χn(A)χk(A−1) = χn−k(A) holds for any invertible n × n matrix A over a field, where χi(A) denotes the coefficient of λn−i in the characteristic polynomial ...
Added: March 9, 2016
Shitov Y., Linear and Multilinear Algebra 2016 Vol. 64 No. 2 P. 219–220
We give a tropical proof of a recent result of Mohindru on the CP-rank conjecture over min-max semiring. ...
Added: December 16, 2015
Shitov Y., / Series math "arxiv.org". 2014. No. 1406.2601.
We construct a nontrivial identity which holds in the semigroup of tropical 3-by-3 matrices. ...
Added: March 14, 2015
Matveenko V., , in: Contemporary MathematicsVol. 616: Tropical and idempotent mathematics and applications.: Providence: American Mathematical Society, 2014. Ch. 11 P. 211–220.
Properties of increasing positively homogeneous functions are studied; in particular, their representations by use of tropical inner products with coefficients chosen from tropical support sets are described. An application to a model of economic complementarityand weak links is developed. It is shown that weak links do not necessary bound total factor productivity from below but ...
Added: February 1, 2015
Providence: American Mathematical Society, 2014.
This volume contains the proceedings of the International Workshop on Tropical and Idempotent Mathematics, held at the Independent University of Moscow, Russia, from August 26-31, 2012. The main purpose of the conference was to bring together and unite researchers and specialists in various areas of tropical and idempotent mathematics and applications. This volume contains articles ...
Added: February 1, 2015
Shitov Y., International Journal of Algebra and Computation 2014 Vol. 24 No. 8 P. 1183–1189
We present a number of new results on the multiplicative structure of univariate polynomials over the Boolean and tropical semirings.We answer the question asked by Kim and Roush in 2005 by proving that almost all Boolean polynomials with nonzero constant term are irreducible.We also give a lower bound for the number of reducible polynomials, and ...
Added: January 8, 2015
Matveenko V., В кн.: Модели и методы тропической математики в прикладных задачах экономики и управленияВып. 2.: СПб.: ВВМ, 2014. Гл. 1 С. 4–23.
Рассматриваются различные варианты CES-функций, получившие распространение в экономике. С точки зрения тропической математики CES-функция без весов интересна тем, что при изменении параметра, связывает идемпотентные операции max и min со стандартной динейно-алгебраической операцией +. Мы рассматриваем CES-функции с позиций тропической математики на основе введения операции обобщенного сложения. Исследуются свойства монотонности, взаимосвязь различных видов CES-функции, а также ...
Added: December 29, 2014
Shitov Y., St Petersburg Mathematical Journal 2014 Vol. 26 No. 2 P. 216–228
The tropical arithmetic operations on R are defined as (a,b) -> min{a,b} and (a,b) -> a+b. We are interested in the concept of a semimodule, which is rather ill-behaved in tropical mathematics. In our paper we study the semimodules S in R^n having topological dimension two, and we show that any such S has always ...
Added: May 24, 2014
M.: Higher School of Economics Publishing House, 2012.
Toric geometry exhibited a profound relation between algebra and topology on one side and combinatorics and convex geometry on the other side. In the last decades, the interplay between algebraic and convex geometry has been explored and used successfully in a much more general setting: first, for varieties with an algebraic group action (such as ...
Added: November 17, 2012