Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / Let $m$ be a positive integer, $p$ be an odd prime, and $\mathbb{Z}_{p^m } = \mathbb{Z}/(p^m )$ be the ring of integers modulo $p^m $. Let
$$Q({\mathbf{x}}) = Q(x_1 ,x_2 ,...,x_n ) = \sum\limits_{1 \leqslant i \leqslant j \leqslant n} {a_{ij} x_i x_j } ,$$
be a quadratic form with integer coefficients. Suppose that $n$ is even and $\det A_Q \not \equiv 0\;(\bmod p)$. Set $\Delta = (( - 1)^{n/2} \det A_Q /p)$, where $( \cdot /p)$ is the Legendre symbol and $\left\| {\mathbf{x}} \right\| = \max \left| {x_i } \right|$. Let $V$ be the set of solutions the congruence
$ $Q({\mathbf{x}})\, \equiv \;0\quad (\bmod p^m ) \quad(1)$$,
contained in $\mathbb{Z}^n $ and let $B$ be any box of points in $\mathbb{Z}^n $of the type
$$B = \left\{ {{\mathbf{x}} \in \mathbb{Z}^n \left| {\,a_i \leqslant x_i < a_i + m_i ,\;\,1 \leqslant i \leqslant n} \right.} \right\},$$
where $a_i ,m_i \in \mathbb{Z},\;1 \leqslant m_i \leqslant p^m $.
In this dissertation we use the method of exponential sums to investigate how large the cardinality of the box $B$ must be in order to guarantee that there exists a solution ${\mathbf{x}}$of (1) in $ B$. In particular we will focus on cubes (all $m_i $equal) centered at the origin in order to obtain primitive solutions with $\left\| {\mathbf{x}} \right\|$ small. For $m = 2$ and $n \geqslant 4$ we obtain a primitive solution with $\left\| {\mathbf{x}} \right\| \leqslant \max \left\{ {2^5 p,2^{18} } \right\}$. For $m = 3$, $n \geqslant 6$, and $\Delta = + 1$, we get $\left\| {\mathbf{x}} \right\| \leqslant \max \left\{ {2^{2/n} p^{(3/2) + (3/n)} ,2^{(2n + 4)/(n - 2)} } \right\}$. Finally for any $m \geqslant 2$, $n \geqslant m,$ and any nonsingular quadratic form we obtain $\left\| {\mathbf{x}} \right\| \leqslant \max \{ 6^{1/n} p^{m[(1/2) + (1/n)]} ,2^{2(n + 1)/(n - 2)} 3^{2/(n - 2)} \} $.
Others results are obtained for boxes $B$ with sides of arbitrary lengths.
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/1631 |
| Date | January 1900 |
| Creators | Hakami, Ali Hafiz Mawdah |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
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