Symmetric Powers of Hyperbolic Forms and of Trace Forms on Symbol Algebras

TL;DR

This paper computes symmetric powers of hyperbolic forms and trace forms on symbol algebras over fields with characteristic not 2, using a generalized Vandermonde convolution to derive explicit results.

Contribution

It provides explicit calculations of symmetric powers for hyperbolic and trace forms on symbol algebras, extending the understanding of their algebraic structure.

Findings

Explicit formulas for symmetric powers of hyperbolic forms

Explicit formulas for symmetric powers of trace forms on symbol algebras

Application of generalized Vandermonde convolution in form computations

Abstract

Let $K$ be a field with characteristic different from 2 and let $S$ be a symbol algebra over $K$. We compute the symmetric powers of hyperbolic quadratic forms over $K$. Also, we compute the symmetric powers of the quadratic trace form of $S$. In both cases we apply a generalised form of the Vandermonde convolution in the course of the computations.