%0 Journal Article
%T Weighted quadratic partitions modulo $P^m$- a new formula and a new demonstration
%A Ali Hafiz Hakami
%J Tamkang Journal of Mathematics
%D 2012
%I Tamkang University
%R 10.5556/j.tkjm.43.2012.11-19
%X Let $Q({f{x}}) = Q(x_1 ,x_2 ,...,x_n )$ be a quadratic form over $mathbb{Z}$, $p$ be an odd prime. Let $V = V_Q = V_{p^m } $ denote the set of zeros of $Q({mathbf{x}})$ in $mathbb{Z}_{p^m }$ and $left| V ight|$ denotes the cardinality of $V$. Set $ phi (V_{p^m } ,{mathbf{y}}) = sum _{{mathbf{x}} in V} e_{p^m } ({mathbf{x}} cdot {mathbf{y}})$ for ${mathbf{y}} e {mathbf{0}}$ and $phi (V_{p^m } ,{mathbf{y}}) = left| {V_{p^m } } ight| - p^{m(n - 1)}$ for ${mathbf{y}} = {mathbf{0}}.$ In this paper we shall give a formula for the calculation of the function $phi (V,{mathbf{y}}).$
%K Quadratic Form
%U http://journals.math.tku.edu.tw/index.php/TKJM/article/view/753