Asymptotic Estimates for the Number of Integer Solutions to Decomposable Form Inequalities

For homogeneous decomposable forms F (X) in n variables with integer coefficients, we consider the number of integer solutions x ∈ Zn to the inequality |F (x)| ≤ m as m → ∞. We give asymptotic estimates which improve on those given previously by the author in [T1]. Here our error terms display desirable behaviour as a function of the height whenever the degree of the form and the number of variables are relatively prime. Introduction In this paper we consider homogeneous polynomials F (X) in n > 1 variables with integer coefficients which factor completely into a product of linear terms over C. Such polynomials are called decomposable forms. We are concerned here with the integer solutions to the Diophantine inequality |F (x)| ≤ m (1) Let V (F ) denote the n-dimensional volume of the set of all real solutions x ∈ R to the inequality |F (x)| ≤ 1, so that by homogeneity mV (F ) is the measure of the set of x ∈ R which satisfy (1). Denote the number of integral solutions to (1) by NF (m). In a previous paper [T1] we answered several open questions regarding (1). For example, NF (m) is finite for all m if and only if F is of finite type: V (F ) is finite, and the same is true for F restricted to any non-trivial subspace defined over Q. Also proven in [T1] was the following asymptotic estimate. [T1, Theorem 3]. Let F be decomposable form of degree d in n variables with integer coefficients. If F is of finite type, then there are a(F ), c(F ) ∈ Q satisfying 1 ≤ a(F ) ≤ d n − 1 n(n−1) and 1991 Mathematics Subject Classification. Primary: 11D75, 11D45; Secondary 11D72.. Research partially supported by NSF grant DMS-0100791 Typeset by AMS-TEX 1 2 JEFFREY LIN THUNDER (d−n) d ≤ c(F ) < ( d n ) (d− n+ 1) such that |NF (m)−mV (F )| ≪ m n−1 d−a(F ) (1 + logm)H(F ) , where the implicit constant depends only on n and d. In particular, |NF (m)−mV (F )| ≪ m n(n−1)2 d(n−1)2+1 (1 + logm)H(F )( d n). The quantity H(F ) appearing here is defined as follows. Write F (X) = ∏d i=1 Li(X) where the Li(X) ∈ C[X] are linear forms in n variables. Denote the coefficient vector of Li(X) by Li and let ‖ · ‖ denote the L norm. Then H(F ) = d