Approximation numbers of linear operators are a very useful tool in order to understand the structure and the numerical behaviour of the operators. In this paper, this concept is extended to polynomials on Banach spaces and the approximation numbers of diagonal polynomials are estimated. As a main tool the rank of polynomials as a graduation of finite type polynomials is introduced and studied.

Dynamical processes with particular reference to opposite phenomena as growth and decay, may be translated into the mathematical language of difference equations or recursive equations. Such equations can be treated in the discrete case(where the principal mathematical instrument is furnished by geometrical progressions) or in the continuous case (where the mathematical instrument is particularly represented by exponential functions and logarithms). The variety of problems and of resolution methods renders the proposed material apt to be used in a significant way in the high school.

For several applications it is very useful to classify the linear or non-linear mappings by their summability properties. Absolutely summing operators and polynomials are prominent and classical examples of such setting. Here we are interested in the larger class of almost summing polynomials and we investigate their connections to other related notions of summability.