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Distribution of roots of polynomial congruences. (English) Zbl 1142.11057
The following problem is dealt with in this paper. For a prime \(p\) and a polynomial \(f(X)\in\mathbb{Z}[X]\), let \[ \mathcal{R}_{p,f}:=\left\{\frac{r}{p}: f(r)\equiv 0 (\mod p), 0\leq r\leq p-1\right\}. \] The author of the article studies the question of how to bound the discrepancy of the set \[ \mathcal{T}_d (p;\mathcal{B})=\left\{\frac{r}{p}\right\}_{r\in\mathcal{R}_{p,f}, f\in\mathcal{F}_d (\mathcal{B}),} \] where \(\mathcal{B}\) is a box defining the range of the coefficients of the (monic) polynomials \(f\) of degree \(d\) in the set \(\mathcal{F}_d (\mathcal{B})\). The main result in the paper gives an upper bound on the discrepancy of \(\mathcal{T}_d(p;\mathcal{B})\) in terms of the cardinality of \(\mathcal{B}\) and \(p\). The bound is non-trivial for sufficiently large boxes \(\mathcal{B}\).
MSC:
11K38 Irregularities of distribution, discrepancy
11J71 Distribution modulo one
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References:
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