zbMATH — the first resource for mathematics

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 \(\mathcal{B}\).
11K38 Irregularities of distribution, discrepancy
11J71 Distribution modulo one
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.