Edge reduction for weakly non-negative quadratic forms

Abstract.

Let $q\colon \Bbb Z^n\to \Bbb Z$ be an integral quadratic form of the shape¶¶$q(x)=\sum\limits\limits ^n_{i=1}q_ix^2_i+\sum\limits\limits _{i\le j}q_{ij}x_ix_j$¶¶with $q_i\le 1$, for every $1\le i\le n$. Several procedures have been introduced to study these forms. In this paper we consider the edge reduction procedure introduced in [7]. ¶A form $q\colon \Bbb Z^n\to \Bbb Z$ is weakly non-negative if $q(x)\ge 0$ for every vector x with non-negative coordinates. Let $q^\prime \colon \Bbb Z^{n+1}\to \Bbb Z$ be obtained from q by edge reduction, then q is weakly non-negative if and only if so is $q^\prime$. We propose an algorithm to decide if q is weakly non-negative.