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Decomposition Statistics of Restricted Polynomial Specializations over Large Finite Fields. (arXiv:1810.07512v1 [math.NT])
来源于:arXiv
For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$
being a prime number) we study the decomposition statistics of its
specializations $$F(t,a_1,\ldots,a_n)\in\mathbf{F}_p[t]$$ with
$(a_1,\ldots,a_n)\in S$, where $S\subset\mathbf{F}_p^n$ is a subset, in the
limit $p\to\infty$ and $\deg F$ fixed. We show that for a sufficiently large
and regular subset $S\subset\mathbf{F}_p^n$, e.g. a product of $n$ intervals of
length $H_1,\ldots,H_n$ with $\prod_{i=1}^nH_n>p^{n-1/2+\epsilon}$, the
decomposition statistics is the same as for unrestricted specializations (i.e.
$S=\mathbf{F}_p^n$) up to a small error. This is a generalization of the
well-known P\'olya-Vinogradov estimate of the number of quadratic residues
modulo $p$ in an interval. 查看全文>>