Some new results on negative polynomial Pell's equation

TL;DR

This paper investigates the existence of polynomial solutions to the negative polynomial Pell's equation for specific classes of polynomials, expanding understanding of solutions in algebraic number theory.

Contribution

It provides new results on the conditions under which polynomial solutions exist for the negative polynomial Pell's equation with certain polynomial classes.

Findings

Identifies classes of polynomials with solutions

Establishes criteria for existence of solutions

Extends previous results in polynomial Pell's equations

Abstract

We consider the negative polynomial Pell's equation $P^2(X)-D(X)Q^2(X)=-1$, where $D(X)\in \mathbb{Z}[X]$ be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions $P(X), \, Q(X)$ with integer coefficients.