A ``class group'' obstruction for the equation \(Cy^d=F(x,z)\)
In this paper, Diophantine equations of the form \[ Cy^d = F(x,z)\tag{1} \] are investigated, where \(F \in \mathbb Z[x,z]\) is an irreducible, primitive form of degree \(n \geq 2\), and \(C \neq 0\), \(d\geq2\) are integers. Let \(K = \mathbb Q (\theta)\) be the algebraic number field, given by \(F(\theta, 1)=0\). The author [C. R., Math., Acad. Sci. Paris 336, No. 1, 7--10 (2003; Zbl 1038.11073)] associates to the form \(F\) an order \(\mathbb Z_F\) within the ring of integers of \(K\), an ideal \(\mathfrak b\) of \(\mathbb Z_F\), and defines the class of \(F\) to be the class of \(\mathfrak b\) within the class group \(Cl(\mathbb Z_F)\). Each proper solution \((x_0, y_0, z_0)\) of (ref {1}) gives rise to an ideal \(\mathfrak D = \mathfrak b (x_0- \theta z_0)\) of \(\mathbb Z_F\) with very special properties, as described in Theorem 3. From this, necessary conditions for the existence of proper solutions of (ref {1}) are derived, in more detail, there must exist some ideal \(\mathfrak c\) of \(\mathbb Z_F\) with norm equal to \(C\) and divisible only by degree \(1\) prime ideals, and the classes of \(\mathfrak c\) and of \(F\) should only differ by a \(d\)-th power inside \(Cl(\mathbb Z_F)\). For \(n=2\) this reduces to previous results of Cassels and Darmon-Granville.
scientific article; zbMATH DE number 5572703
scientific article