Polynomial Values in Subfields and Affine Subspaces of Finite Fields

baran stanley j.;

For an integer r, a prime power q and a polynomial f over a finite field Fqr of qr 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 Fqr . We also obtain similar results for elements in affine subspaces of
Fqr , considered as a linear space over Fq .