Cipra, James Arthur2010-05-122010-05-122010-05-12http://hdl.handle.net/2097/4152This thesis establishes bounds (primarily upper bounds) on Waring's number in finite fields. Let $p$ be a prime, $q=p^n$, $\mathbb F_q$ be the finite field in $q$ elements, $k$ be a positive integer with $k|(q-1)$ and $t= (q-1)/k$. Let $A_k$ denote the set of $k$-th powers in $\mathbb F_q$ and $A_k' = A_k \cap \mathbb F_p$. Waring's number $\gamma(k,q)$ is the smallest positive integer $s$ such that every element of $\mathbb F_q$ can be expressed as a sum of $s$ $k$-th powers. For prime fields $\mathbb F_p$ we prove that for any positive integer $r$ there is a constant $C(r)$ such that $\gamma(k,p) \le C(r) k^{1/r}$ provided that $\phi(t) \ge r$. We also obtain the lower bound $\gamma(k,p) \ge \frac {(t-1)}ek^{1/(t-1)}-t+1$ for $t$ prime. For general finite fields we establish the following upper bounds whenever $\gamma(k,q)$ exists: $$ \gamma(k,q)\le 7.3n\left\lceil\frac{(2k)^{1/n}}{|A_k^\prime|-1}\right\rceil\log(k), $$ $$ \gamma(k,q)\le8n \left\lceil{\frac{(k+1)^{1/n}-1}{|A^\prime_k|-1}}\right\rceil, $$ and $$ \gamma(k,q)\ll n(k+1)^{\frac{\log(4)}{n\log|\kp|}}\log\log(k). $$ We also establish the following versions of the Heilbronn conjectures for general finite fields. For any $\ve>0$ there is a constant, $c(\ve)$, such that if $|A^\prime_k|\ge4^{\frac{2}{\ve n}}$, then $\gamma(k,q)\le c(\varepsilon) k^{\varepsilon}$. Next, if $n\ge3$ and $\gamma(k,q)$ exists, then $\gamma(k,q)\le 10\sqrt{k+1}$. For $n=2$, we have $\gamma(k,p^2)\le16\sqrt{k+1}$.en-US© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/Waring's ProblemNumber TheoryDissertationMathematics (0405)