Sums of four rational squares with certain restrictions

TL;DR

This paper proves a four-square theorem for nonnegative rational numbers with additional linear restrictions, and explores conjectures about representing rationals with mixed powers and quadratic forms.

Contribution

It establishes a new four-square theorem for rational numbers under linear restrictions and proposes several conjectures on representing rationals with mixed powers.

Findings

Every nonnegative rational can be expressed as four rational squares with a linear restriction.

The paper introduces conjectures on representing rationals as sums involving fourth powers and squares.

Provides a foundation for further research on rational representations with algebraic restrictions.

Abstract

In this paper we mainly study sums of four rational squares with certain restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any $a,b,c,d\in\mathbb Q_{\ge0}$, each $r\in\mathbb Q_{\ge0}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Q_{\ge0}$ such that $ax+by+cz+dw$ is a rational square (or a rational cube). This paper also contains many conjectures; for example, for any positive integers $a$ and $b$ with $\gcd(a,b)=1$, we conjecture that each $r\in\mathbb Q_{\ge0}$ can be written as $aw^4+bx^4+y^2+z^2$ with $w,x,y,z\in\mathbb Q$.