WEYL'S INEQUALITY AND SYSTEMS OF FORMS

Let F1, …, Fr be integer forms of degree d ≥ 2 in s variables. Relaxing the non-singularity condition in a well-known result by Birch, we establish the expected Hardy–Littlewood asymptotic formula for the density of integer points on the intersection F1 = ··· = Fr = 0, providing that $$\begin{equation} s \gt \max_{\textbf{a} \in \mathbb{Z}^r \backslash \{\boldsymbol{0}\}} \dim V_{\textbf{a}}^* + r(r+1) (d-1) 2^{d-1},\end{equation}$$ where Va^* = {x ∈ ℂ^s: ∇(a1 F1(x) + ··· + ar Fr(x)) = 0}. In the same context, we also improve on previous work by Schmidt and show that the expected Hardy–Littlewood asymptotic formula holds true, providing that each form in the rational pencil of F1, …, Fr has rank exceeding 2r² + 2r (d = 2); h-invariant exceeding 8r² + 8r (d = 3); h-invariant exceeding φ(d) (d − 1) 2^d−1 r(r + 1) for a certain function φ(d) when d≥4. In particular, if F1, …, Fr are rational quadratic forms, each form in their complex pencil has rank exceeding 2r² + 2r, and the intersection F1 = ··· = Fr = 0 has a non-singular real zero, then this intersection also has a non-trivial rational zero. For r = 1, this recovers a classical result by Meyer.Our new tool, which is of interest in itself, is a variant of Weyl's inequality for general systems of forms that is useful in situations such as those above where one knows a certain lower rank (or dimension of singular locus) bound for all forms in the rational pencil of the given ones.