Leseprobe
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An Introduction to G-Functions. (AM-133), Volume 133
Francis J. Sullivan, Giovanni Gerotto, Bernard Dwork, et al.
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s.
After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, André, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and André on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.
Kundenbewertungen
Limit of a sequence, Cauchy sequence, Algebraic variety, Commutative ring, Density theorem, Parameter, Riemann hypothesis, Linear differential equation, Laurent series, Galois group, Identity matrix, Siegel's lemma, Polynomial, Rational number, Algebraic closure, Automorphism, Analytic function, Theorem, Special case, Separable polynomial, Cohomology, Algebraically closed field, Elliptic curve, Formal power series, Generic point, Hypergeometric function, Summation, Meromorphic function, Root of unity, P-adic number, Fuchs' theorem, Cauchy's theorem (geometry), Residue field, Weil conjecture, Equation, G-module, Ring of integers, Coefficient, Number theory, General linear group, Mathematical induction, Analytic continuation, Radius of convergence, Complete intersection, Topology of uniform convergence, Algebraic Method, Eigenvalues and eigenvectors, Valuation ring, Natural number, Projective line, Differential equation, Binomial series, Monotonic function, Triangle inequality, Geometry, Argument principle, Dimension (vector space), Field of fractions, Existential quantification, Newton polygon, Galois extension, Algebraic number field, Monodromy, Multiplicative group, Exponential function, Finite field, Exterior algebra, Discrete valuation, Subring, Quadratic residue