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- W2982596092 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." @default.
- W2982596092 created "2019-11-08" @default.
- W2982596092 creator A5038301210 @default.
- W2982596092 date "2019-10-31" @default.
- W2982596092 modified "2023-09-26" @default.
- W2982596092 title "Secondary Terms in Asymptotics for the Number of Zeros of Quadratic Forms" @default.
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