Class numbers of binary quadratic polynomials

TL;DR

This paper derives a formula for the proper class number of binary quadratic polynomials under certain divisibility conditions, enabling analysis of their growth and finiteness properties.

Contribution

It introduces a new formula for proper class numbers of binary quadratic polynomials and explores their growth and finiteness under specific divisibility assumptions.

Findings

Derived a formula for proper class numbers with divisibility conditions.

Studied the growth of class numbers for totally positive binary quadratic polynomials.

Proved finiteness results for polynomials with fixed quadratic part and class number.

Abstract

In this paper, we give a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of totally positive binary quadratic polynomials. As an application, we prove finiteness results on totally positive binary quadratic polynomials with a fixed quadratic part and a fixed proper class number.