Let be a univariate polynomial of degree n with real or complex coefficients.

For physical systems, and are polynomials in with real coefficients:

A quadratic equation with real or complex coefficients has two solutions, called roots.

It can be proven that there exists no equation of the form in which a, b and c are all real coefficients.

The polynomials of the transfer function will all have real coefficients.

In practice if (1) has real coefficients, the solutions based on y will have very small imaginary parts which must be discarded.

First, if the matrix is real, then its characteristic equation must have real coefficients.

One approach to construct regular functions is to use power series with real coefficients.

It was originally designed and has been further developed to be particularly suited to polynomials with real, random coefficients.

Consequently, if is a polynomial with real coefficients, and , then as well.