Representations of integral quadratic polynomials

TL;DR

This paper investigates the classification of integral quadratic polynomials, establishing finiteness results for certain classes of universal and regular forms, and extends the discussion to more general algebraic settings.

Contribution

It proves finiteness of equivalence classes for positive ternary universal quadratic polynomials and regular ternary triangular forms, and generalizes to Dedekind domains within global fields.

Findings

Finitely many equivalence classes of positive ternary universal integral quadratic polynomials.

Finitely many regular ternary triangular forms.

Extension of the theory to Dedekind domains in global fields.

Abstract

In this paper, we study the representations of integral quadratic polynomials. Particularly, it is shown that there are only finitely many equivalence classes of positive ternary universal integral quadratic polynomials, and that there are only finitely many regular ternary triangular forms. A more general discussion of integral quadratic polynomials over a Dedekind domain inside a global field is also given.