Polynomial Values in Subfields and Affine Subspaces of Finite Fields

Oliver Roche-Newton, Igor Shparlinski

Published 2014-07-08, updated 2014-07-26Version 2

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 ${\mathbb F}_q$.