On a general Thue's equation

Abstract

We analyze the integral points on varieties defined by one equation of the form f 1 · · · f r = g , where the f i , g are polynomials in n variables with algebraic coefficients, and g has "small" degree; we shall use a method that we recently introduced in the context of Siegel's Theorem for integral points on curves. Classical, very particular, instances of our equations arise, e.g., in a well-known corollary of Roth's Theorem (the case n = 2, f i linear forms, deg g < r -2) and with the norm-form equations , treated by W. M. Schmidt. Here we shall prove (Thm. 1) that the integral points are not Zariski-dense, provided Σdeg f i > n · max (deg f i ) + deg g and provided the f i , g , satisfy certain (mild) assumptions which are "generically" verified. Our conclusions also cover certain complete-intersection subvarieties of our hypersurface (Thm. 2). Finally, we shall prove (Thm. 3) an analogue of the Schmidt's Subspace Theorem for arbitrary polynomials in place of linear forms.