Pell equations for polynomials
type: Item
instance of: scholarly article
title: Pell equations for polynomials
zbMATH Open document ID: 0894.11008
DOI: 10.1016/S0019-3577(97)81817-4
author: Fumio Hazama
published in: Indagationes Mathematicae. New Series
publication date: 1998-09-08T00:00:00Z
review text: Let \(k\) be a field of characteristic \(\neq 2\) and \(k^*= k-\{0\}\). For \(d\in k^*\), we denote by \(C_d\) the conic defined by the equation \(y^2= x^2-d\). The function field \(k(C_d)\) is a quadratic extension of \(k(x)\). Then the twist \(\widetilde{C}_e\) of another conic \(C_e\) \((e\in k^*)\) by the extension \(k(C_d)/k(x)\) is defined over \(k(x)\) by the equation \[ (x^2-d)Y^2= X^2-e. \] In section 2 of this paper the author proves that the twisted conic \(\widetilde{C}_e\) has a solution in polynomials of degree \(n\) if and only if \(d\equiv e\pmod{(k^*)^2}\) when \(n\) is odd and \(e\equiv 1 \pmod{(k^*)^2}\) when \(n\) is even. Furthermore, under this condition there is only one solution for each degree \(n\) up to sign. Let \(f(x)\in k[x]\) be a square-free polynomial. In section 3, the results of section 2 and those of \textit{F. Hazama} [J. Number Theory 37, 168-172 (1991; Zbl 0729.14019)] are generalized and a necessary and sufficient condition is obtained for the equation \(X^2-f(x)Y^2= 1\) to be solvable in polynomials. Finally, in the last section the notion of primitive solution is introduced and a sufficient condition for it is given.
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 1136778
zbMATH Keywords: generalized Jacobian variety
zbMATH Keywords: quadratic diophantine equations
zbMATH Keywords: Pell equations for polynomials
zbMATH Keywords: twisted conic
zbMATH Keywords: primitive solution
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
cites work: On the Mordell-Weil group of certain abelian varieties defined over the rational function field
cites work: Polynomial Pell's Equations
cites work: Q4039754
cites work: Q4320820
Recommended article: Q4869119
Recommended article: On the solvability of equations of higher degree
Recommended article: A characterization of linear polynomials
Recommended article: ADDITIVE POLYNOMIALS AND PRIMITIVE ROOTS OVER FINITE FIELDS
Recommended article: Q4519459
Recommended article: The coefficients of primitive polynomials over finite fields
Recommended article: Mordell's equation in characteristic three
Recommended article: Solving conics over functions fields
Recommended article: Function fields of conics as invariant subfields
Recommended article: Q3211445
description: scientific article; zbMATH DE number 1136778