Integral points of small height outside of a hypersurface.pdf

a r X i v : m a t h / 0 4 0 9 3 7 4 v 3 [ m a t h .N T ] 3 N o v 2 0 0 5 INTEGRAL POINTS OF SMALL HEIGHT OUTSIDE OF A HYPERSURFACE LENNY FUKSHANSKY Abstract. Let F be a non-zero polynomial with integer coefficients in N variables of degree M . We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem. 1. Introduction and notation Let F (X) = F (X1, ..., XN ) ∈ Z[X1, ..., XN ] be a polynomial in N ≥ 1 variables of degree M ≥ 1 with integer coefficients. If F is not identically zero, there must exist a point with integer coordinates at which F does not vanish, in other words an integral point that lies outside of the hypersurface defined by F over Q. How does one find such a point? For a point x = (x1, ..., xN ) ∈ ZN , define its height and length respectively by H(x) = max 1≤i≤N |xi|, L(x) = N∑ i=1 |xi|. It is easy to see that a set of points with height or length bounded by some fixed constant is finite. In fact, for a positive real number R, (1) ∣∣{x ∈ ZN : H(x) ≤ R}∣∣ = (2[R] + 1)N , and (2) ∣∣{x ∈ ZN : L(x) ≤ R}∣∣ = min([R],N)∑ k=0 2k ( N k )( [R] k ) , where (1) is obvious and (2) follows from Theorem 6 of [4]; we write [R] for the integer part of R. Therefore if we were able to prove the existence of a point x ∈ ZN with H(x) ≤ R or L(x) ≤ R for some explicitly determined value of R, then the problem of finding this point would reduce to a finite search. Thus we will consider the following problem. 1991 Mathematics Subject Classification. Primary 11C08, 11H06; Secondary 11D04, 11H46. Key words and phrases. polynomials, lattices, linear forms, height. 1 2