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20
Analytification is the limit of all tropicalizations
"... Abstract. We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings. 1. ..."
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Abstract. We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings. 1.
The combinatorics of frieze patterns and Markoff numbers
, 2007
"... ... model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky’s cluster algebras of type A. One can derive from the matchings model an enumerativ ..."
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... model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky’s cluster algebras of type A. One can derive from the matchings model an enumerative meaning for the Markoff numbers, and prove that the associated Laurent polynomials have positive coefficients as was conjectured (much more generally) by Fomin and Zelevinsky. Most of this research was conducted under the auspices of REACH (Research Experiences in Algebraic Combinatorics at Harvard).
Monomial Maps and Algebraic Entropy
 TO BE SUBMITTED TO ERGODIC THEORY AND DYNAMICAL SYSTEMS
, 2006
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MODULI SPACES OF METRIC GRAPHS OF GENUS 1 WITH MARKS ON VERTICES
, 2008
"... In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces MGv 1,n, which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of t ..."
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In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces MGv 1,n, which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the socalled scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.
WORKING WITH TROPICAL MEROMORPHIC FUNCTIONS OF ONE VARIABLE
"... Abstract. In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For instance, we introduce maximally represented trop ..."
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Abstract. In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For instance, we introduce maximally represented tropical polynomials and tropical polynomials in compact forms to quickly find roots of given tropical polynomials. We also prove the existence and uniqueness of tropical theorems for meromorphic functions with prescribed roots and poles. Moreover, we explain the relations between classical and tropical meromorphic functions. Different definitions and applications of tropical meromorphic functions are discussed. Finally, we point out the properties of tropical meromorphic functions are very similar to complex ones and prove some tropical analogues of theorems in complex analysis. 1.
LatticeBased Minimum Error Rate Training using Weighted FiniteState Transducers with Tropical Polynomial Weights
"... Minimum Error Rate Training (MERT) is a method for training the parameters of a loglinear model. One advantage of this method of training is that it can use the large number of hypotheses encoded in a translation lattice as training data. We demonstrate that the MERT line optimisation can be modelle ..."
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Minimum Error Rate Training (MERT) is a method for training the parameters of a loglinear model. One advantage of this method of training is that it can use the large number of hypotheses encoded in a translation lattice as training data. We demonstrate that the MERT line optimisation can be modelled as computing the shortest distance in a weighted finitestate transducer using a tropical polynomial semiring. 1
A Glimpse at Supertropical Valuation theory
"... We give a short tour through major parts of a recent long paper [IKR1] on supertropical valuation theory, leaving aside nearly all proofs (to be found in [IKR1]). In this way we hope to give easy access to ideas of a new branch of so called “supertropical algebra”. 1 ..."
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We give a short tour through major parts of a recent long paper [IKR1] on supertropical valuation theory, leaving aside nearly all proofs (to be found in [IKR1]). In this way we hope to give easy access to ideas of a new branch of so called “supertropical algebra”. 1
DEGREEGROWTH OF MONOMIAL MAPS
"... ABSTRACT. For projectivizations of rational maps Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective ..."
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ABSTRACT. For projectivizations of rational maps Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.