Analytic Methods for Diophantine Problems.

Analytic Number Theory;Circle Method;Mathematics;Science;Mathematics

Spencer, Craig ValereMilicevic, Djordje ;

University of Michigan

关键词: Analytic Number Theory;    Circle Method;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/60823/cvspenc_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis studies applications of the circle method to various Diophantine problems.In particular, we explore the following four themes.First, we develop the Bentkus-G;;otze-Freeman variant of the Davenport-Heilbronn method for function fields in order to count $mathbb{F}_q[t]$-solutions to diagonal Diophantine inequalities in $mathbb{F}_q((1/t)).$Suppose that $k$ and $s$ are natural numbers with $k>1,$$$s ge frac{4}{3}k(log k+log log k)+O(k),$$and $ch (mathbb{F}_q)nmid k.$Let $tau$ be a fixed integer, and let $lambda_1,ldots,lambda_s$ be fixed non-zero elements of $mathbb{F}_q ((1/t)),$ not all in $mathbb{F}_q(t)$-rational ratio.Suppose also that the equation$$lambda_1z_1^k+cdots+lambda_sz_s^k=0$$has a non-trivial solution $mathbf{z}$ in $mathbb{F}_q((1/t))^s.$Then, for all sufficiently large positive real numbers $P$, the number of $mathbb{F}_q[t]$-solutions $N(P;boldsymbol{lambda})$ of $$ord (lambda_1x_1^k+cdots+lambda_sx_s^k)