Geodesic
MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS
Glasgow mathematical journal, Tome 60 (2018) no. 2, pp. 487-493

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DOI

We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J. 139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers. 
SHPARLINSKI, IGOR E. MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS. Glasgow mathematical journal, Tome 60 (2018) no. 2, pp. 487-493. doi: 10.1017/S0017089517000222
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     title = {MULTIPLICATIVE {ORDERS} {IN} {ORBITS} {OF} {POLYNOMIALS} {OVER} {FINITE} {FIELDS}},
     journal = {Glasgow mathematical journal},
     pages = {487--493},
     year = {2018},
     volume = {60},
     number = {2},
     doi = {10.1017/S0017089517000222},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000222/}
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