A Ternary Algebra with Applications to Binary Quadratic Forms

TL;DR

This paper explores a ternary algebra structure related to binary quadratic forms, providing new multiplicative formulas and a generalized multiplication for points on conic sections, extending classical algebraic concepts.

Contribution

It introduces a ternary algebra framework for binary quadratic forms and derives new multiplicative formulas and point multiplication methods on conics.

Findings

Ring forms a ternary algebra under a triple product

Derived multiplicative formulas for quadratic forms

Generalized multiplication for points on conic sections

Abstract

We discuss multiplicative properties of the binary quadratic form $a x^2 + b x y + c y^2$ by considering a ring of matrices which is closed under a triple product. We prove that the ring forms a ternary algebra in the sense of Hestenes, and then derive both multiplicative formulas for a large class of binary quadratic forms and a type of multiplication for points on a conic section which generalizes the algebra of rational points on the unit circle.