Hakami, Ali Hafiz Mawdah2009-08-032009-08-032009-08-03http://hdl.handle.net/2097/1631Let $m$ 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.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/Small solutionsQuadratic formsSmall solutions of quadratic congruences modulo p^mQuadratic congruencesSmall zerosDissertationMathematics (0405)