Pell equations for polynomials
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.
scientific article; zbMATH DE number 1136778
scientific article