Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

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FOLDING EQUILATERAL PLANE GRAPHS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195913600017 ◽

2013 ◽

Vol 23 (02) ◽

pp. 75-92 ◽

Cited By ~ 1

Author(s):

ZACHARY ABEL ◽

ERIK D. DEMAINE ◽

MARTIN L. DEMAINE ◽

SARAH EISENSTAT ◽

JAYSON LYNCH ◽

...

Keyword(s):

Plane Graph ◽

Analogous Problem ◽

Central Vertex ◽

Polyhedral Complex ◽

Plane Graphs ◽

Single Vertex ◽

Np Completeness ◽

Np Complete ◽

Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

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The Viro method for construction of Bernstein-Bézier algebraic hypersurface piece

Science China Mathematics ◽

10.1007/s11425-012-4409-8 ◽

2012 ◽

Vol 55 (6) ◽

pp. 1269-1279 ◽

Cited By ~ 5

Author(s):

YiSheng Lai ◽

WeiPing Du ◽

RenHong Wang

Keyword(s):

Algebraic Hypersurface ◽

Viro Method

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Piecewise Affine Dynamical Models of Petri Nets – Application to Emergency Call Centers*

Fundamenta Informaticae ◽

10.3233/fi-2021-2086 ◽

2022 ◽

Vol 183 (3-4) ◽

pp. 169-201

Author(s):

Xavier Allamigeon ◽

Marin Boyet ◽

Stéphane Gaubert

Keyword(s):

Petri Nets ◽

Call Centers ◽

Nonexpansive Mappings ◽

Emergency Call ◽

Free Fluid ◽

Periodic Regime ◽

Timed Petri Nets ◽

Polyhedral Complex ◽

Piecewise Affine ◽

Paris Area

We study timed Petri nets, with preselection and priority routing. We represent the behavior of these systems by piecewise affine dynamical systems. We use tools from the theory of nonexpansive mappings to analyze these systems. We establish an equivalence theorem between priority-free fluid timed Petri nets and semi-Markov decision processes, from which we derive the convergence to a periodic regime and the polynomial-time computability of the throughput. More generally, we develop an approach inspired by tropical geometry, characterizing the congestion phases as the cells of a polyhedral complex. We illustrate these results by a current application to the performance evaluation of emergency call centers in the Paris area. We show that priorities can lead to a paradoxical behavior: in certain regimes, the throughput of the most prioritary task may not be an increasing function of the resources.

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Zero Counting and Invariant Sets of Differential Equations

International Mathematics Research Notices ◽

10.1093/imrn/rnx199 ◽

2017 ◽

Vol 2019 (13) ◽

pp. 4119-4158

Author(s):

Gal Binyamini

Keyword(s):

Vector Field ◽

Differential Equations ◽

Upper Bound ◽

Linear Differential Equations ◽

Invariant Sets ◽

Polynomial Vector ◽

Algebraic Hypersurface ◽

Polynomial Differential Equations ◽

Polynomial Vector Field ◽

Linear Differential

Abstract Consider a polynomial vector field $\xi$ in ${\mathbb C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of degree $d$. We introduce a condition on $\xi$ called constructible orbits and show that under this condition $N(K,d)$ grows polynomially with $d$. We establish the constructible orbits condition for linear differential equations over ${\mathbb C}(t)$, for planar polynomial differential equations and for some differential equations related to the automorphic $j$-function. As an application of the main result, we prove a polylogarithmic upper bound for the number of rational points of a given height in planar projections of $K$ following works of Bombieri–Pila and Masser.

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The Viro Method for Construction of Piecewise Algebraic Hypersurfaces

Abstract and Applied Analysis ◽

10.1155/2013/690341 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Yisheng Lai ◽

Weiping Du ◽

Renhong Wang

Keyword(s):

New Method ◽

Algebraic Hypersurface ◽

And Topology ◽

Algebraic Hypersurfaces ◽

Viro Method

We propose a new method to construct a real piecewise algebraic hypersurface of a given degree with a prescribed smoothness and topology. The method is based on the smooth blending theory and the Viro method for construction of Bernstein-Bézier algebraic hypersurface piece on a simplex.

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