Let s(n, q) be the smallest number s such that any n-fold Fq-valued
interpolation problem in Pk
Fq has a solution of degree s, that is: For any
pairwise different Fq-rational points P1, . . . , Pn there exists a hypersurface
H of degree s defined over Fq such that P1, . . . , Pn−1 ∈ H and Pn 6∈ H.
This function s(n, q) was studied by Ernst Kunz and the second author
in [KuW] and completely determined for q = 2 and q = 3. For q ≥ 4, we
improve the results from [KuW].
The affine analogue to s(n, q) is the smallest number s = sa(n, q) such
that any n-fold Fq-valued interpolation problem in Ak(Fq), k ∈ N>0 has
a polynomial solution of degree ≤ s. We exactly determine this number.