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.
General
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