Symmetric homogeneous diophantine equations of odd degree

TL;DR

This paper develops a method to find rational parametric solutions for symmetric homogeneous diophantine equations of odd degree, generalizing from specific cases to broader classes with explicit parameter counts.

Contribution

It introduces a parametric solution approach for symmetric forms of any odd degree, extending previous solutions from 5th degree to all odd degrees n ≥ 5.

Findings

Parametric solutions exist for symmetric forms of odd degree in specified variable counts.

The number of parameters depends linearly on the degree n.

The approach applies to equations of the form F(x_i)=q, with explicit parameter counts.

Abstract

We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the following results. (1) Every symmetric form of odd degree $n\ge 5$ in $6 \cdot 2^{n-5}$ variables has a rational parametric solution depending on $2n-8$ parameters. (2) Let $F(x_1, ..., x_N)$ be a symmetric form of odd degree $n\ge 5$ in $N=6 \cdot 2^{n-4}$ variables, and let $q$ be any rational number. Then the equation $F(x_i)=q$ has a rational parametric solution depending on $2n-6$ parameters. The latter result can be viewed as a solution of a problem of Waring type for this class of forms.