The representation of integers by positive ternary quadratic polynomials

TL;DR

This paper proves that for any fixed conductor, there are finitely many equivalence classes of positive primitive ternary regular complete quadratic polynomials, extending previous finiteness results in quadratic form theory.

Contribution

It establishes a finiteness theorem for positive primitive ternary regular complete quadratic polynomials with a fixed conductor, generalizing earlier results for quadratic forms and triangular forms.

Findings

Finiteness of equivalence classes for fixed conductor c

Extension of Watson's finiteness results to quadratic polynomials

Generalization of Chan and Oh's results for ternary triangular forms

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 ${\mathbf x} = (x_1, \ldots, x_n)$ 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.