A polynomial function is a function whose values can be expressed in terms of a defining polynomial.
A polynomial function of maximum degree 0 is said to be a constant function, while a polynomial function with maximum degree 1 is called a linear function.
Named polynomial functions include named orthogonal polynomials such as the Hermite polynomial and Legendre polynomial.
Over an infinite field, a polynomial in one variable is determined by its polynomial function. Over a finite
field 
, this need not hold: 
and 
induce the same function. Every
function 
has a unique representative polynomial
of polynomial degree less than 
. Indeed, polynomial
interpolation gives such a representative, and the difference of two representatives
of polynomial degree less than 
cannot have 
distinct roots unless
it is zero. Thus the polynomial degree of an
arbitrary representative need not be an intrinsic property of its function
(Shapiro 2015, p. 14).
