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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
##### Keywords:
Polynomial congruences; discrepancy
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##### References:
 [1] C. Hooley, “On the distribution of the roots of polynomial congruences,” Mathematika, vol. 11, pp. 39-49, 1964. · Zbl 0123.25802 · doi:10.1112/S0025579300003466 [2] C. Hooley, “On the greatest prime factor of a quadratic polynomial,” Acta Mathematica, vol. 117, no. 1, pp. 281-299, 1967. · Zbl 0146.05704 · doi:10.1007/BF02395047 [3] W. Duke, J. B. Friedlander, and H. Iwaniec, “Equidistribution of roots of a quadratic congruence to prime moduli,” Annals of Mathematics, vol. 141, no. 2, pp. 423-441, 1995. · Zbl 0840.11003 · doi:10.2307/2118527 [4] Á. Tóth, “Roots of quadratic congruences,” International Mathematics Research Notices, vol. 2000, no. 14, pp. 719-739, 2000. · Zbl 1134.11339 · doi:10.1155/S1073792800000404 [5] H. Weyl, “Zur Abschätzung von \zeta (1+it),” Annals of Mathematics, vol. 10, pp. 88-101, 1921. · JFM 48.0346.01 [6] H. Iwaniec and E. Kowalski, Analytic Number Theory, vol. 53 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2004. · Zbl 1059.11001 [7] B. Poonen and J. F. Voloch, “Random Diophantine equations,” in Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, Calif, 2002), vol. 226 of Progr. Math., pp. 175-184, Birkhäuser, Boston, Mass, USA, 2004. · Zbl 1208.11050 [8] E. Bombieri, “On exponential sums in finite fields,” American Journal of Mathematics, vol. 88, no. 1, pp. 71-105, 1966. · Zbl 0171.41504 · doi:10.2307/2373048 [9] A. Granville, I. E. Shparlinski, and A. Zaharescu, “On the distribution of rational functions along a curve over Fp and residue races,” Journal of Number Theory, vol. 112, no. 2, pp. 216-237, 2005. · Zbl 1068.11043 · doi:10.1016/j.jnt.2005.02.002
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