On a general Thue's equation

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 Sigmadeg f(i) > n . max (deg f(i)) + deg g and provided the fi, g, satisfy certain (mild) assumptions which are "generically" verified. Our conclusions also cover certain complete-intersection subvarieties of our hypersurface (Thm. 2). FinaIly, we shall prove (Thm. 3) an analogue of the Schmidt's Subspace Theorem for arbitrary polynomials in place of linear forms.