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Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.

512 --- Algebra --- Geometry, algebraic --- Tropical geometry --- Study and teaching.

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"For a connected smooth projective curve of genus g, global sections of any line bundle L with deg(L) 2g 1 give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry: We replace projective space by tropical projective space, and an embedding by a homeomorphism onto its image preserving integral structures (or equivalently, since is a curve, an isometry), which is called a faithful tropicalization. Let be an algebraically closed field which is complete with respect to a nontrivial nonarchimedean value. Suppose that is defined over and has genus g 2 and that is a skeleton (that is allowed to have ends) of the analytification an of in the sense of Berkovich. We show that if deg(L) 3g 1, then global sections of L give a faithful tropicalization of into tropical projective space. As an application, when Y is a suitable affine curve, we describe the analytification Y an as the limit of tropicalizations of an effectively bounded degree"--

Geometry, Algebraic. --- Tropical geometry. --- Algebraic geometry -- Tropical geometry -- Tropical geometry. --- Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Rigid analytic geometry. --- Algebraic geometry -- Cycles and subschemes -- Divisors, linear systems, invertible sheaves.

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This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.

Algebraic geometry. --- Algebraic Geometry. --- Geometry, Enumerative. --- Logarithms. --- Tropical geometry. --- Geometria enumerativa --- Geometria tropical --- Logaritmes

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Mathematical statistics --- Geometry, Algebraic --- Algebraic geometry -- Instructional exposition (textbooks, tutorial papers, etc.) --- Algebraic geometry -- Real algebraic and real analytic geometry -- Semialgebraic sets and related spaces. --- Algebraic geometry -- Special varieties -- Determinantal varieties. --- Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra. --- Algebraic geometry -- Tropical geometry -- Tropical geometry. --- Biology and other natural sciences -- Genetics and population dynamics -- Problems related to evolution. --- Commutative algebra -- Computational aspects and applications -- GrÃjabner bases; other bases for ideals and modules (e.g., Janet and border bases) --- Convex and discrete geometry -- Polytopes and polyhedra -- Lattice polytopes (including relations with commutative algebra and algebraic geometry) --- Operations research, mathematical programming -- Mathematical programming -- Integer programming. --- Probability theory and stochastic processes -- Markov processes -- Markov chains (discrete-time Markov processes on discrete state spaces) --- Statistics -- Instructional exposition (textbooks, tutorial papers, etc.) --- Statistics -- Multivariate analysis -- Contingency tables. --- Statistics -- Parametric inference -- Hypothesis testing. --- Commutative algebra -- Computational aspects and applications -- Solving polynomial systems; resultants.