Polynomial function

In mathematics, a polynomial function is a function obtained by evaluating a polynomial.

By language abuse, a polynomial function is sometimes called a polynomial, thus confusing the notion of polynomial function with that of polynomial. This confusion is not serious in the context of polynomials with real or complex coefficients (or more generally with coefficients in a field) but can lead to misinterpretations in more general contexts (for example for polynomials with coefficients in a finite field).

Real or complexe polynomial functions

An exemple of a real polynomial function of degree 5

A common case is where the polynomial has real or complex coefficients. More precisely, consider the polynomial P of the form

$P=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$

where the a_j are real or complex numbers. The associated polynomial function f is then defined by

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}$

where the argument x can be either real or complex.

The most common examples are:

• the constant functions corresponding to n = 0;
• the linear functions corresponding to n = 1;
• the quadratic functions n = 2;
• the cubic functions corresponding to n = 3.

The degree of a real or complex polynomial function is defined as the degree of the polynomial with which it is associated (with the convention that the degree is -∞ if the function is constant).

Since a non-constant real or complex polynomial of degree n has at most n roots according to the fundamental theorem of algebra, one deduces that a real or complexe polynomial function that is non-constant has at most n zeros. In other words, two real or complex polynomial functions of degree less than or equal to n that coincide on more than n points are necessarily identical (that is, they have the same degree and same coefficients).

A real or complex polynomial function f is infinitely derivable and the k-th derivative of f is exactly the polynomial function associated with the k-th formal derivative of P. For example, the formal derivative of P is given by

$P^{\prime }=na_{n}X^{n-1}+(n-1)a_{n-1}X^{n-2}+\cdots +2a_{2}X+a_{1}$

and one has

$f^{\prime }(x)=na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots +2a_{2}x+a_{1}.$

In particular, the derivatives of order k > n of polynomial functions of degree n are necessarily identically zero.

Similarly, the primitives of f are exactly the polynomial functions associated with the formal primitives of P, that is, of the form

$x\mapsto {\frac {a_{n}}{n+1}}x^{n+1}+{\frac {a_{n-1}}{n}}x^{n}+\cdots +{\frac {a_{1}}{2}}x^{2}+a_{0}x+C$

where C is a real or arbitrary complex constant.

Polynomial function with coefficients in a field

More generally, it is possible to consider a polynomial P with coefficients in a field K:

$P=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$

where the a_j are elements of K. The associated polynomial function f is then the function from K to itself defined by

$\forall x\in K,\quad f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}.$

In the case where the field is infinite (for example in the case of real numbers or complex numbers considered above), one can still identify polynomials and polynomial functions. In other words, the application that maps a polynomial with coefficients in K to the corresponding polynomial function is injective from the set of polynomials with coefficients in K to the set of applications from K to itself. Therefore, one can define as in the previous section the degree of a polynomial function as the degree of the corresponding polynomial, and two polynomial functions are identical if and only if their associated polynomials are the same.

In the case where K is a finite field, the foregoing is no longer true. For instance, in the field with two elements, the polynomial X(X - 1) is not the null polynomial but the associated polynomial function is identically zero. It is then not possible to define the notion of degree of a polynomial function and two polynomial functions can be identical without their associated polynomials being equal. This shows that it is necessary in this context to distinguish the notion of polynomial function with that of polynomial.

Polynomial functions and associative algebra

One can also consider a polynomial P with coefficients in a ring A:

$P=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$

where the a_j are elements of A. One can as above defines the associated polynomial function.

More generally, it is possible to consider an associative algebra E over A and the function that maps an element e of E to the element P(e) of E defined by

$P(e)=a_{n}e^{n}+a_{n-1}e^{n-1}+\cdots +a_{1}e+a_{0}\,1_{E}.$

This map is a ring homomorphism.

A very common case of use is where A is a field K and where E is the set of square matrices with coefficients in K of a given size. If M is such a matrix, one has

$P(M)=a_{n}M^{n}+a_{n-1}M^{n-1}+\cdots +a_{1}M+a_{0}\,\mathrm {Id}$

where Id is the identity matrix. Thus, for any polynomial P and any square matrix M, P(M) is a matrix. The notion of matrix polynomial plays a central role a linear algebra, for instance to diagonalize or triangularize matrices.