The arithmetic of polynomials over a finite field possesses deep and fascinating parallels with classical number theory. This volume develops all the tools needed to provide modern, "adelic" proofs of the polynomial analogs to two of the most famous theorems of classical additive number theory: Vinogradov's 3-Primes Theorem and the Waring Problem. Series: Oxford Mathematical Monographs. Num Pages: 174 pages, bibliography. BIC Classification: PBF; PBH. Category: (P) Professional & Vocational; (UP) Postgraduate, Research & Scholarly. Dimension: 238 x 163 x 15. Weight in Grams: 416. . 1991. hardcover. . . . .

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Bibliographic Details

Title: Additive Number Theory of Polynomials over a Finite Field (Oxford Mathematical Monographs)
Publisher: Clarendon Press
Publication Date: 1991
Language: English
ISBN 10: 019853583X
ISBN 13: 9780198535836
Binding: Hardcover
Condition: New

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Synopsis

This volume is a systematic treatment of the additive number theory of polynomials over a finite field, an area possessing deep and fascinating parallels with classical number theory. In providing asymptomatic proofs of both the Polynomial Three Primes Problem (an analog of Vinogradov's theorem) and the Polynomial Waring Problem, the book develops the various tools necessary to apply an adelic "circle method" to a wide variety of additive problems in both the polynomial and classical settings. A key to the methods employed here is that the generalized Riemann hypothesis is valid in this polynomial setting. The authors presuppose a familiarity with algebra and number theory as might be gained from the first two years of graduate course, but otherwise the book is self-contained. Starting with analysis on local fields, the main technical results are all proved in detail so that there are extensive discussions of the theory of characters in a non-Archimidean field, adele class groups, the global singular series and Radon-Nikodyn derivatives, L-functions of Dirichlet type, and K-ideles.

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'To sum up, the book is an outstanding motivation for and introduction to the study of analytical methods in arithmetic.' mededelingen van het wiskundig genootschap, no.8, - november 1993

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