Secondary terms in asymptotics for the number of zeros of quadratic forms

Abstract: Let $F$ be a non-degenerate quadratic form on an $n$-dimensional vector space $V$ over the rational numbers. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size $B$, roughly speaking. For example, Heath-Brown gave an asymptotic of the form: $c_1 B^{n-2} +O_{J,\epsilon, \omega}(B^{(n-1)/2+\epsilon})$, for any $\epsilon > 0$ and dim$V \geq 5$, where $c_1 \in \mathbb{C}$ and $\omega \in \mathcal{S}(V(\mathbb{R}))$ is a smooth function. More recently, Getz gave an asymptotic of the form: $c_1 B^{n-2} + c_2 B^{n/2}+O_{J,\epsilon, \omega}(B^{n/2+\epsilon-1})$ when $n$ is even, in which $c_2 \in \mathbb{C}$ has a pleasant geometric interpretation. We consider the case where $n$ is odd and give an analogous asymptotic of the form: $c_1 B^{n-2} +c_2B^{(n-1)/2}+O_{J,\epsilon,\omega}(B^{n/2+\epsilon-1})$. Notably it turns out that the geometric interpretation of the constant $c_2$ of the asymptotic in the odd degree and even degree cases is strikingly different.