Additive Number Theory of Polynomials over a Finite Field

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.

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This volume is a systematic treatment of the additive number theory of polynomials over a finite field, an area possessing fascinating and deep parallels with classical number theory. In providing asymptotic 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 Generalised 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 a graduate course, but otherwise the book is self-contained.