On the concentration of points of polynomial maps and applications

Cilleruelo, Javier; Garaev, M. Z.; Ostafe, Alina; Shparlinski, Igor E.

doi:10.1007/s00209-011-0959-7

Cited by 37 publications

(72 citation statements)

References 12 publications

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“…It is certainly a natural question to extend the results and methods of [4][5][6][7] to the case of segments of trajectories in subgroups.…”

Section: Commentsmentioning

confidence: 98%

Groups generated by iterations of polynomials over finite fields

Shparlinski¹

2015

Proceedings of the Edinburgh Mathematical Society

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Given a finite field Fq of q elements, we consider a trajectory of the map u → f (u) associated with a polynomial f ∈ Fq[X]. Using bounds of character sums, under some mild condition on f , we show that for an appropriate constant C > 0 no N Cq 1/2 distinct consecutive elements of such a trajectory are contained in a small subgroup G of F * q , improving the trivial lower bound #G N . Using a different technique, we also obtain a similar result for very small values of N . These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.

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“…It is certainly a natural question to extend the results and methods of [4][5][6][7] to the case of segments of trajectories in subgroups.…”

Section: Commentsmentioning

confidence: 98%

Groups generated by iterations of polynomials over finite fields

Shparlinski¹

2015

Proceedings of the Edinburgh Mathematical Society

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“…The method is related to several recent results on the distribution of solutions to polynomial congruences in very small boxes, see [30][31][32] for further references. Both the bound of [32] on I (R, S; M) and some estimates of [30][31][32] on the density of solutions of polynomial congruences have been improved and generalised in [28].…”

Section: Isogeny and Isomorphism Classes In Various Familiesmentioning

confidence: 98%

“…Using the method of [30], which in turn has been further developed in [31], in [32] a lower bound of an asymptotically correct order of magnitude is given on I (R, S; M). The method is related to several recent results on the distribution of solutions to polynomial congruences in very small boxes, see [30][31][32] for further references.…”

Section: Isogeny and Isomorphism Classes In Various Familiesmentioning

confidence: 99%

Elliptic Curves over Finite Fields: Number Theoretic and Cryptographic Aspects

Shparlinski

2013

Advances in Applied Mathematics, Modeling, and Computational Science

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“…In this situation, Weil's theorem on exponential sums permits proving equidistribution of the partial orbit. For M p 1/2 , Weil's theorem becomes inapplicable and lower bounds on diam O z,M based on Vinogradov's theorem were established in [6].…”

Section: Introductionmentioning

confidence: 99%