POLYNOMIAL VALUES IN SUBFIELDS AND AFFINE SUBSPACES OF FINITE FIELDS
Oliver Roche‐Newton, Igor E. Shparlinski · The Quarterly Journal of Mathematics · 2014
For an integer |$r$|, a prime power |$q$| and a polynomial |$f$| over a finite field |${{\mathbb F}}_{q^r}$| of |$q^r$| elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of |$f$| which fall in a proper subfield of |${{\mathbb F}}_{q^r}$|. We also obtain similar results for elements in affine subspaces of |${{\mathbb F}}_{q^r}$|, considered as a linear space over |${{\mathbb F}}_q$|.