An 
-polynomial over a finite
field 
of even order 
is a polynomial 
of degree at most 
inducing a permutation of
, with 
and 
, such that for every 
the map
 |
(1)
|
is also a permutation of 
. This is the normalized convention. For example, 
is an 
-polynomial.
The points 
, for 
, together with 
and 
form a hyperoval in the
projective plane over 
. This relates these polynomials to deep
holes of dimension-two Reed-Solomon codes
(Chen and Huang 2026).
