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Abstract: Let $q,Q$ be two integral quadratic forms in $m < n$ variables. One can ask when $q$ can be represented by $Q$ - that is, whether there exists an $n times m$-integer matrix $T$ such that $Q circ T = q$. Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo $N$ for every positive integer $N$. In the absence of local obstructions, does a (global) representation of $q$ by $Q$ exist?