Polynomial Bounds for the Solutions of a Class of Diophantine Equations Original Research Article

LetKbe an algebraic number field such that all the embeddings ofKinto image are real. We denote byOKthe ring of algebraic integers ofK. LetF(X, Y) be an irreducible polynomial inK[X, Y]−K[Y] of total degreeNand of degreen>0 inY. We denote byFN(X, Y) its leading homogeneous part. Suppose thatFN(1, Y) is a polynomial of degreenhaving no real roots. In this paper we establish a polynomial upper bound for the size of solutions (x, y)set membership, variantOK×Kof the equationF(X, Y)=0.