On algebraic approximations of certain algebraic numbers

Abstract

Using hypergeometric functions and the Thue–Siegel method we give an effective improvement of Liouville's approximation theorem. As an application, we derive effective upper bounds for the solutions ( X, Y) of the two-parametric family of quartic Thue inequalities |BX 4−AX 3Y−6BX 2Y 2+AXY 3+BY 4|⩽N for A⩾58| B| 3 or A⩾308 B 4.