Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / 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:
$$
\gamma(k,q)\le
7.3n\left\lceil\frac{(2k)^{1/n}}{|A_k^\prime|-1}\right\rceil\log(k),
$$
$$
\gamma(k,q)\le8n \left\lceil{\frac{(k+1)^{1/n}-1}{|A^\prime_k|-1}}\right\rceil,
$$
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 $\gamma(k,p^2)\le16\sqrt{k+1}$.
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/4152 |
| Date | January 1900 |
| Creators | Cipra, James Arthur |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
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