# Waring’s number in finite fields

## K-REx Repository

 dc.contributor.author Cipra, James Arthur dc.date.accessioned 2010-05-12T13:06:11Z dc.date.available 2010-05-12T13:06:11Z dc.date.issued 2010-05-12T13:06:11Z dc.identifier.uri http://hdl.handle.net/2097/4152 dc.description.abstract This 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: en_US $$\gamma(k,q)\le 7.3n\left\lceil\frac{(2k)^{1/n}}{|A_k^\prime|-1}\right\rceil\log(k),$$ $$\gamma(k,q)\le8n \left\lceil{\frac{(k+1)^{1/n}-1}{|A^\prime_k|-1}}\right\rceil,$$ 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 $\gamma(k,p^2)\le16\sqrt{k+1}$. dc.language.iso en_US en_US dc.publisher Kansas State University en dc.subject Waring's Problem en_US dc.subject Number Theory en_US dc.title Waring’s number in finite fields en_US dc.type Dissertation en_US dc.description.degree Doctor of Philosophy en_US dc.description.level Doctoral en_US dc.description.department Department of Mathematics en_US dc.description.advisor Todd E. Cochrane en_US dc.subject.umi Mathematics (0405) en_US dc.date.published 2010 en_US dc.date.graduationmonth May en_US
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