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.

Integral points on curves: Siegel’s theorem after Siegel’s proof

“…7 This was later extended by P. Corvaja and the second author in [15] and further by J.-H. Evertse and R. Ferretti [25], again using the Subspace Theorem. In the meantime J.-H. Evertse, and independently A. van der Poorten and H.P.…”

Essentially large divisors and their arithmetic and function-theoretic inequalities

“…In this direction, we first discovered that the filtration method from [CZ04a] (see also [Ru04]) yields the expected sharp bound (even for the original version of largeness) in the case of X = P q as stated in the following theorem. (Cf.…”

“…Proofs of the sharp sufficient criteria for essentially large divisors. As was stated in the Introduction, the filtration method from [CZ04a], [Ru04] can be used to prove the sharp bound for the number of components of a large divisor in the case of X = P q (Theorem 1.7). We now give the proof.…”

“…In [CZ02], Corvaja-Zannier found an innovative way of using Schmidt's Subspace Theorem to give a new proof of the classical Theorem of Siegel on integral points on affine curves. They subsequently expanded their approach to obtain certain results on integral points in higher dimensions ([CZ03], [CZ04a], [CZ04b], [CZ06a], [CZ06b]). The approach was translated to Nevanlinna theory in [Ru04].…”

The Second Main Theorem for Hypersurfaces

“…In this section we recall Corvaja and Zannier's filtration as made more explicit in [3]. Details of proofs can be found in [3], and also [2].…”