The representation of integers by positive ternary quadratic polynomials

Abstract

An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form $Q(\mathbf{x}+\mathbf{v})$, where $Q$ is an integral quadratic form in the variables $\mathbf{x}=(x_1,\dots ,x_n)$ and $\mathbf{v}$ is a vector in ${\mathbb{Q}}^n$. Its conductor is defined to be the smallest positive integer $c$ such that $c\mathbf{v}\in {\mathbb{Z}}^n$. We prove that for a fixed positive integer $c$, there are only finitely many equivalence classes of positive primitive ternary regular complete quadratic polynomials with conductor $c$. This generalizes the analogous finiteness results for positive definite regular ternary quadratic forms by Watson and for ternary triangular forms by Chan and Oh.