Vito Volterra (1860-1940) was one of the founding fathers of functional analysis. At the turn of the twentieth century, he introduced the notion of *functions of lines*, which are defined over a functional space. He studied their derivative and obtained results on integral equations. Practically, a Volterra series is a polynomial functional expansion similar to a Taylor series that provides an approximation of weakly nonlinear systems. One of the first application to nonlinear system analysis is due to Wiener in the 1940s, who developed a method for determining the nonlinear response to a white noise input. Nowadays, this approach is widely used for system identification in many domains such as electrical engineering or biological sciences.

# Polynomials

# Division by increasing power order

The division of polynomials by increasing power order is similar to the usual Euclidean division but in reversed order : the terms of lower degree of the dividend are eliminated first. Some useful algorithms can be deduced from this process, such as series expansion or partial fraction decomposition.

**EUCLIDEAN DIVISION**

It is well known that given two polynomials and on a commutative field K, it is always possible to perform the Euclidean division