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30
Nonarchimedean amoebas and tropical varieties
, 2004
"... We study the nonarchimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using nonarchimedean analysis a ..."
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Cited by 79 (0 self)
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We study the nonarchimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using nonarchimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the localglobal principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.
Tannakian duality for AndersonDrinfeld motives and algebraic independence of Carlitz logarithms
 Invent. Math
"... Abstract. We develop a theory of Tannakian Galois groups for tmotives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given tmotive is equal to the dimension of its Galois group. Using this resul ..."
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Cited by 26 (6 self)
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Abstract. We develop a theory of Tannakian Galois groups for tmotives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given tmotive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent. Contents
The algebraic closure of the power series field in positive characteristic
 PROC. AMER. MATH. SOC
, 2001
"... For K an algebraically closed field, let K((t)) denote the quotient field of the power series ring over K. The “NewtonPuiseux theorem ” states that if K has characteristic 0, the algebraic closure of K((t)) is the union of the fields K((t 1/n)) over n ∈ N. We answer a question of Abhyankar by con ..."
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Cited by 19 (5 self)
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For K an algebraically closed field, let K((t)) denote the quotient field of the power series ring over K. The “NewtonPuiseux theorem ” states that if K has characteristic 0, the algebraic closure of K((t)) is the union of the fields K((t 1/n)) over n ∈ N. We answer a question of Abhyankar by constructing an algebraic closure of K((t)) for any field K of positive characteristic explicitly in terms of certain generalized power series.
Fibers of tropicalization
 Math. Z
"... Abstract. We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained ..."
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Cited by 12 (2 self)
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Abstract. We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained “constructively” using Kedlaya’s transfinite version of Newton’s method. 1.
Measure theory and integration on the LeviCivita field
 Contemporary Mathematics
"... Abstract. It is well known that the disconnectedness of a nonArchimedean totally ordered field in the order topology makes integration more difficult than in the real case. In this paper, we present a remedy to that difficulty and study measure theory and integration on the LeviCivita field. After ..."
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Cited by 8 (4 self)
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Abstract. It is well known that the disconnectedness of a nonArchimedean totally ordered field in the order topology makes integration more difficult than in the real case. In this paper, we present a remedy to that difficulty and study measure theory and integration on the LeviCivita field. After reviewing basic elements of calculus on the field, we introduce a measure that proves to be a natural generalization of the Lebesgue measure on the field of the real numbers and have similar properties. Then we introduce a family of simple functions from which we obtain a larger family of measurable functions and derive a simple characterization of such functions. We study the properties of measurable functions, we show how to integrate them over measurable sets of R, and we show that the resulting integral satisfies similar properties to those of the Lebesgue integral of real calculus. 1.
Analytical properties of power series on LeviCivita fields
 Ann. Math. Blaise Pascal
, 2005
"... A detailed study of power series on the LeviCivita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are i ..."
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Cited by 8 (4 self)
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A detailed study of power series on the LeviCivita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and reexpandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are infinitely often differentiable. 1
OneDimensional Optimization on NonArchimedean Fields. Journal of Nonlinear and Convex Analysis, 2:351–361
, 2001
"... Abstract. One dimensional optimization on nonArchimedean fields is presented. We derive first and second order necessary and sufficient optimality conditions. For first order optimization, these conditions are similar to the corresponding real ones; but this is not the case for higher order optimiz ..."
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Cited by 7 (5 self)
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Abstract. One dimensional optimization on nonArchimedean fields is presented. We derive first and second order necessary and sufficient optimality conditions. For first order optimization, these conditions are similar to the corresponding real ones; but this is not the case for higher order optimization. This is due to the total disconnectedness of the given nonArchimedean field in the order topology, which renders the usual concept of differentiability weak. We circumvent this difficulty by using a stronger concept of differentiability based on the derivate approach, which entails a Taylor formula with remainder and hence a similar local behavior as in the real case. 1.
On the topological structure of the LeviCivita field
 J. Math. Anal. Appl
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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Cited by 5 (2 self)
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are
Power series and padic algebraic closures
 Journal of Number Theory
"... We describe a presentation of the algebraic closure of the ring of Witt vectors of an algebraically closed field of characteristic p> 0. The construction uses “generalized power series in p ” as constructed by Poonen, based on an example of Lampert. 1 ..."
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Cited by 4 (1 self)
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We describe a presentation of the algebraic closure of the ring of Witt vectors of an algebraically closed field of characteristic p> 0. The construction uses “generalized power series in p ” as constructed by Poonen, based on an example of Lampert. 1