Determination of a binary quadratic form by its values at integer points

Abstract

Let f = f(x, y) be a quadratic form with real coefficients in two integer variables x, y. Let V(f) be the set of values taken by f(x, y) at points (x, y) ≠ (0,0). Impose the same conditions on a second form f′. Trivially, f equivalent to f′ implies V(f) = V(f′). It will be shown that the converse implication holds in general for definite forms; the obvious exception f = x² + xy + y², f′ = x² + 3y² will be shown to be essentially the only one.