Quartic Equations with Trivial Solutions over Gaussian Integers

TL;DR

This paper investigates specific quartic equations over Gaussian integers using resolvent methods, providing new proofs for known theorems and analyzing equations like Mordell's over Gaussian integers.

Contribution

It introduces new equations involving quartic forms over Gaussian integers and offers novel proofs for existing theorems in this domain.

Findings

New proofs for equations like $X^4+Y^4=Z^2$ and variants over Gaussian integers.

Analysis of equations such as $X^4+6X^2 Y^2+Y^4=Z^2$ and $X^4\pm Y^4=(1+i)Z^2$.

Extension of classical results to Gaussian integer solutions.

Abstract

In our work we study the equations of the form $aX^4+bX^2 Y^2+cY^4=dZ^2$ over Gaussian integers by a method of the resolvents. We study as a new equations $X^4+6X^2 Y^2+Y^4=Z^2$ (Mordell's equation over $\mathbb{Z}[i]$), $X^4+6(1+i)X^2Y^2+2iY^4=Z^2$ and $X^4\pm Y^4=(1+ i)Z^2$ and give the new proofs of the known theorems on $X^4+Y^4=Z^2$ (Fermat - Hilbert), $X^4\pm Y^4=iZ^2$ (Szab\'o - Najman).