Let
f be a function from a finite field
with a prime number
p of elements, to
. In this article we consider those functions
f(
X) for which there is a positive integer
with the property that
f(
X)
i, when considered as an element of
, has degree at most
p−2−
n+
i, for all
i=1,…,
n. We prove that every line is incident with at most
t−1 points of the graph of
f, or at least
n+4−
t points, where
t is a positive integer satisfying
n>(
p−1)/
t+
t−3 if
n is even and
n>(
p−3)/
t+
t−2 if
n is odd. With the additional hypothesis that there are
t−1 lines that are incident with at least
t points of the graph of
f, we prove that the graph of
f is contained in these
t−1 lines. We conjecture that the graph of
f is contained in an algebraic curve of degree
t−1 and prove the conjecture for
t=2 and
t=3. These results apply to functions that determine less than
directions. In particular, the proof of the conjecture for
t=2 and
t=3 gives new proofs of the result of Lovász and Schrijver [L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Sci. Math. Hungar. 16 (1981) 449–454] and the result in [A. Gács, On a generalization of Rédei’s theorem, Combinatorica 23 (2003) 585–598] respectively, which classify all functions which determine at most 2(
p−1)/3 directions.
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