Tropical Curves, Convex Domains, Sandpiles and Amoebas
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DESCRIBING AMOEBAS 1. Introduction
ArXiV.org/1805.00273 June 2021 Pacific J. Math, to appear. DESCRIBING AMOEBAS MOUNIR NISSE AND FRANK SOTTILE Abstract. An amoeba is the image of a subvariety of an algebraic torus under the loga- rithmic moment map. We consider some qualitative aspects of amoebas, establishing results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semi-algebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba which is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semi-algebraic sets. 1. Introduction Hilbert’s basis theorem implies that an algebraic variety is the intersection of finitely many hypersurfaces. A tropical variety is the intersection of finitely many tropical hyper- surfaces [5, 10]. These results not only provide a finite description of classical and tropical varieties, but they are important algorithmically, for they allow these varieties to be repre- sented and manipulated on a computer. Amoebas and coamoebas are other objects from tropical geometry that are intermediate between classical and tropical varieties, but less is known about how they may be represented. The amoeba of a subvariety V of a torus (C×)n is its image under the logarithmic moment map (C×)n Rn [8, Ch. 6]. The coamoeba is the set of arguments in V . An early study of amoebas→ [19] discussed their computation. Purbhoo [16] showed that the amoeba of a variety V is the intersection of amoebas of all hypersurfaces containing V and that points in the complement of its amoeba are witnessed by certain lopsided polynomials in its ideal. -
"The Newton Polygon of a Planar Singular Curve and Its Subdivision"
THE NEWTON POLYGON OF A PLANAR SINGULAR CURVE AND ITS SUBDIVISION NIKITA KALININ Abstract. 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 ∆ of the curve C. If a given point p is of multiplicity m on C, then the coefficients of F are subject to certain linear constraints. These constraints can be visualized in the above subdivision of ∆. Namely, we find a 3 2 distinguished collection of faces of the above subdivision, with total area at least 8 m . The union of these faces can be considered to be the “region of influence” of the singular point p in the subdivision of ∆. We also discuss three different definitions of a tropical point of multiplicity m. 0. Introduction Fix a non-empty finite subset Z2 and any valuation field K. We consider a curve C given by an equation F (x,y) = 0, where A⊂ i j ∗ (1) F (x,y)= aijx y , aij K . ∈ (i,jX)∈A Suppose that we know only the valuations of the coefficients of the polynomial F (x,y). Is it possible to extract any meaningful information from this knowledge? Unexpectedly, many geometric properties of C are visible from such a viewpoint. The Newton polygon ∆=∆( ) of the curve C is the convex hull of in R2. The extended Newton A A2 polyhedron of the curve C is the convex hull of the set ((i, j),s) R R (i, j) ,s val(aij) . -
Dissipation in the Abelian Sandpile Model
Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Faculteit Technische Natuurwetenschappen Delft Institute of Applied Mathematics Dissipation in the Abelian Sandpile Model Verslag ten behoeve van het Delft Institute of Applied Mathematics als onderdeel ter verkrijging van de graad van BACHELOR OF SCIENCE in Technische Wiskunde en Technische Natuurkunde door HENK JONGBLOED Delft, Nederland Augustus 2016 Copyright c 2016 door Henk Jongbloed. Alle rechten voorbehouden. BSc verslag TECHNISCHE WISKUNDE en TECHNISCHE NATUURKUNDE \Dissipation in the Abelian Sandpile Model" HENK JONGBLOED Technische Universiteit Delft Begeleiders Prof. dr.ir. F.H.J. Redig Dr.ir. J.M. Thijssen Overige commissieleden Dr. T. Idema Dr. J.L.A. Dubbeldam Dr. J.A.M. de Groot Augustus, 2016 Delft Abstract The Abelian Sandpile model was originally introduced by Bak, Tang and Wiesenfeld in 1987 as a paradigm for self-organized criticality. In this thesis, we study a variant of this model, both from the point of view of mathematics, as well as from the point of view of physics. The effect of dissipation and creation of mass is investigated. By linking the avalanche dynamics of the infinite-volume sandpile model to random walks, we derive some criteria on the amount of dissipation and creation of mass in order for the model to be critical or non-critical. As an example we prove that a finite amount of conservative sites on a totally dissipative lattice is not critical, and more generally, if the distance to a dissipative site is uniformly bounded from above, then the model is not critical. We apply also applied a renormalisation method to the model in order to deduce its critical exponents and to determine whether a constant bulk dissipation destroys critical behaviour. -
Abelian Sandpile Model on Symmetric Graphs
Abelian Sandpile Model on Symmetric Graphs Natalie J. Durgin Francis Edward Su, Advisor David Perkinson, Reader May, 2009 Department of Mathematics Copyright c 2009 Natalie J. Durgin. The author grants Harvey Mudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copy- ing is by permission of the author. To disseminate otherwise or to republish re- quires written permission from the author. Abstract The abelian sandpile model, or chip-firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self- organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sand- pile group is isomorphic to the symmetric subgroup of the original graph. These explorations are motivated by possible applications to counting the domino tilings of a 2n × 2n grid. Contents Abstract iii Acknowledgments ix 1 Introduction to the Theory of Sandpiles 1 1.1 Origins of the Model . .1 1.2 Model Structure and Definitions . .3 1.3 Standard Sandpile Theorems . .7 1.4 Sandpile Literature . 13 1.5 Alternate Approaches to Standard Theorems . 14 2 Symmetric Sandpiles 19 2.1 Applications to Tiling Theorems . 19 2.2 Graphs with Symmetry . 21 2.3 Summary of Notation . 25 2.4 The Symmetric Sandpile Subgroup . 26 3 Developing a Quotient Graph 29 3.1 The Row Quotient . -
The Sandpile Load-Balancing Problem for Latency Minimization
THE SANDPILE LOAD BALANCER AND LATENCY MINIMIZATION PROBLEM BERJ CHILINGIRIAN MASSACHUSETTS INSTITUTE OF TECHNOLOGY Abstract. I present the sandpile load balancer, an application of the abelian sandpile model to dynamic load balancing. The sandpile load balancer exploits the self-organized criticality of the abelian sandpile model to distribute web traffic across a network of servers. I also propose the sandpile load balancer latency minimization problem in which the goal is to minimize the latency of packets across the network of servers. I show the number of possible strategies for solving this problem and explore heuristic-based approaches via simulation. 1. Introduction Bak, Tang, and Wiesenfeld introduced the abelian sandpile model in 1987. Their seminal paper was concerned with modeling the avalanches that occur in growing piles of sand and how the frequency of such avalanches relates to pink noise [1]. Dhar extended their work in 1990 by proving several mathematical properties of the abelian sandpile model [2]. One such property is self-organized criticality: the tendency of the system to converge on a critical state in which it is simultaneously stable and on the edge of chaos. Coincidentally, self-organized criticality has also been observed in a variety of dynamic systems in nature [1]. The notion of self-organized criticality conveniently applies to dynamic load bal- ancing in which a network of servers distribute incoming traffic in order to \balance" the processing burden. This paper is concerned with the application of the abelian sand pile model to dynamic load balancing in order to exploit self-organized criti- cality for a real-world problem. -
Harmonic Dynamics of the Abelian Sandpile
Harmonic dynamics of the abelian sandpile Moritz Langa,1 and Mikhail Shkolnikova aInstitute of Science and Technology Austria, 3400 Klosterneuburg, Austria Edited by Yuval Peres, Microsoft Research, Redmond, WA, and approved January 3, 2019 (received for review July 12, 2018) The abelian sandpile is a cellular automaton which serves as “avalanche.” Due to the loss of particles at the boundaries of the the archetypical model to study self-organized criticality, a phe- domain, this process eventually terminates (ref. 4, theorem 1), nomenon occurring in various biological, physical, and social pro- and the “relaxed” sandpile reaches a stable configuration which cesses. Its recurrent configurations form an abelian group, whose is independent of the order of topplings (ref. 3, p. 13). The distri- identity is a fractal composed of self-similar patches. Here, we ana- bution of avalanche sizes—the total number of topplings after a lyze the evolution of the sandpile identity under harmonic fields random particle drop—follows a power law (1) and is thus scale of different orders. We show that this evolution corresponds to invariant. However, the critical exponent for this power law is yet periodic cycles through the abelian group characterized by the unknown (5). smooth transformation and apparent conservation of the patches Soon after the introduction of the sandpile model (1), it was constituting the identity. The dynamics induced by second- and observed that the set of stable configurations can be divided third-order harmonics resemble smooth stretchings and transla- into two classes, recurrent and transient ones (6). Thereby, a tions, respectively,while the ones induced by fourth-order harmon- stable configuration is recurrent if it appears infinitely often ics resemble magnifications and rotations. -
Abelian Sandpile Model on Symmetric Graphs Natalie Durgin Harvey Mudd College
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Scholarship@Claremont Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2009 Abelian Sandpile Model on Symmetric Graphs Natalie Durgin Harvey Mudd College Recommended Citation Durgin, Natalie, "Abelian Sandpile Model on Symmetric Graphs" (2009). HMC Senior Theses. 217. https://scholarship.claremont.edu/hmc_theses/217 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Abelian Sandpile Model on Symmetric Graphs Natalie J. Durgin Francis Edward Su, Advisor David Perkinson, Reader May, 2009 Department of Mathematics Copyright c 2009 Natalie J. Durgin. The author grants Harvey Mudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copy- ing is by permission of the author. To disseminate otherwise or to republish re- quires written permission from the author. Abstract The abelian sandpile model, or chip-firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self- organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sand- pile group is isomorphic to the symmetric subgroup of the original graph. -
Kepler's Area Law in the Principia: Filling in Some Details in Newton's
Historia Mathematica 30 (2003) 441–456 www.elsevier.com/locate/hm Kepler’s area law in the Principia: filling in some details in Newton’s proof of Proposition 1 Michael Nauenberg Department of Physics, University of California, Santa Cruz, CA 95064, USA Abstract During the past 30 years there has been controversy regarding the adequacy of Newton’s proof of Prop. 1 in Book 1 of the Principia. This proposition is of central importance because its proof of Kepler’s area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the Principia.Itis shown here that the critics of Prop. 1 have misunderstood Newton’s continuum limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well-grounded. 2003 Elsevier Inc. All rights reserved. Résumé Au cours des 30 dernières années, il y a eu une controverse au sujet de la preuve de la Proposition 1 telle qu’elle est formulée par Newton dans le premier livre de ses Principia. Cette proposition est d’une importance majeure puisque la preuve qu’elle donne de la loi des aires de Kepler permit à Newton d’introduire une expression géometrique du temps, lui permettant ainsi de résoudre des problèmes dans la domaine de la dynamique orbitale. Nous démontrons ici que les critiques de la Proposition 1 ont mal compris l’argument de Newton relatif aux limites continues et qu’ils ont négligé de considérer la justification pour ces limites donnée par Newton dans son Lemme 3. -
Absolute Irreducibility of Polynomials Via Newton Polytopes
ABSOLUTE IRREDUCIBILITY OF POLYNOMIALS VIA NEWTON POLYTOPES SHUHONG GAO DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634 USA [email protected] Abstract. A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein’s crite- rion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbi- trarily in the field, but keeping certain collection of them nonzero. 1. Introduction It is well-known that Eisenstein’s criterion gives a simple condition for a polyno- mial to be irreducible. Over the years this criterion has witnessed many variations and generalizations using Newton polygons, prime ideals and valuations; see for examples [3, 25, 28, 38]. We examine the Newton polygon method and general- ize it through Newton polytopes associated with multivariable polynomials. This leads us to a more general geometric criterion for absolute irreducibility of multi- variable polynomials. Since the Newton polygon of a polynomial is only a small fraction of its Newton polytope, our method is much more powerful. Absolute irre- ducibility of polynomials is crucial in many applications including but not limited to finite geometry [14], combinatorics [47], algebraic geometric codes [45], permuta- tion polynomials [23] and function field sieve [1]. We present many infinite families of absolutely irreducible polynomials over an arbitrary field. These polynomials remain absolutely irreducible even if their coefficients are modified arbitrarily but with certain collection of them nonzero. -
Characterizing and Tuning Exceptional Points Using Newton Polygons
Characterizing and Tuning Exceptional Points Using Newton Polygons Rimika Jaiswal,1 Ayan Banerjee,2 and Awadhesh Narayan2, ∗ 1Undergraduate Programme, Indian Institute of Science, Bangalore 560012, India 2Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India (Dated: August 3, 2021) The study of non-Hermitian degeneracies { called exceptional points { has become an exciting frontier at the crossroads of optics, photonics, acoustics and quantum physics. Here, we introduce the Newton polygon method as a general algebraic framework for characterizing and tuning excep- tional points, and develop its connection to Puiseux expansions. We propose and illustrate how the Newton polygon method can enable the prediction of higher-order exceptional points, using a recently experimentally realized optical system. As an application of our framework, we show the presence of tunable exceptional points of various orders in PT -symmetric one-dimensional models. We further extend our method to study exceptional points in higher number of variables and demon- strate that it can reveal rich anisotropic behaviour around such degeneracies. Our work provides an analytic recipe to understand and tune exceptional physics. Introduction{ Energy non-conserving and dissipative Isaac Newton, in 1676, in his letters to Oldenburg and systems are described by non-Hermitian Hamiltoni- Leibniz [54]. They are conventionally used in algebraic ans [1]. Unlike their Hermitian counterparts, they are not geometry to prove the closure of fields [55] and are in- always diagonalizable and can become defective at some timately connected to Puiseux series { a generalization unique points in their parameter space { called excep- of the usual power series to negative and fractional ex- tional points (EPs) { where both the eigenvalues and the ponents [56, 57]. -
Newton Polygons
Last revised 12:55 p.m. October 21, 2018 Newton polygons Bill Casselman University of British Columbia [email protected] Newton polygons are associated to polynomials with coefficients in a discrete valuation ring, and they give information about the valuations of roots. There are several applications, among them to the structure of Dieudonne´ modules, the ramification of local field extensions, and the desingularization of algebraic curves in P2. As an exception to a common practice of attribution, Newton polygons were originally introduced by Isaac Newton himself, and how he used them is not so different from how they are used now. He wanted to solve polynomial equations f(x, y)=0 for y as a series in fractional powers of x. For example, to solve yn = x we write simply y = x1/n, and 1/n to solve yn =1+ x set y = xk . X0 k Newton sketched the procedure he had come up with in a letter to Oldenburg. It is quite readable (see p. 126 of [Newton:1959] for the original Latin, p. 145 for an English translation). The diagram he exhibits is not essentially different from the ones drawn today. Newton polygons are perfectly and appropriately named—they are non•trivial, and introduced by Newton. This thread became eventually a method related to desingularizing algebraic curves over C (explained in Chaper IV of [Walker:1950]). My original motivation in writing this essay was to understand the theory of crystals, particularly 5 of § Chapter IV of [Demazure:1970]. But since then I have come across other applications. -
Finsler Metric, Systolic Inequality, Isometr
DISCRETE SURFACES WITH LENGTH AND AREA AND MINIMAL FILLINGS OF THE CIRCLE MARCOS COSSARINI Abstract. We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian sur- faces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n−1) 2 squares. We prove that our discrete FAC is equivalent to the FAC for surfaces with self-reverse metric. If the surface is a disk, the discrete FAC follows from Steinitz's algo- rithm for transforming curves into pseudolines. This gives a combinato- rial proof of the FAC for disks with self-reverse metric. We also imitate Ivanov's proof of the same fact, using discrete differential forms. And we prove a new theorem: the FAC holds for M¨obiusbands with self- reverse metric. To prove this we use a combinatorial curve shortening flow developed by de Graaf-Schrijver and Hass-Scott. The same method yields a proof of the systolic inequality for Klein bottles with self-reverse metric, conjectured by Sabourau{Yassine. Self-reverse metrics can be discretized using walls because every normed plane satisfies Crofton's formula: the length of every segment equals the symplectic measure of the set of lines that it crosses.