A soft introduction to algebraic entropy

The goal of this mainly expository paper is to develop the theory of the algebraic entropy in the basic setting of vector spaces V over a field K. Many complications encountered in more general settings do not appear at this first level. We will prove the basic properties of the algebraic entropy of linear transformations ϕ:V→V of vector spaces and its characterization as the rank of V viewed as module over the polynomial ring K[X] through the action of ϕ. The two main theorems on the algebraic entropy, namely, the Addition Theorem and the Uniqueness Theorem, whose proofs require many efforts in more general settings, are easily deduced from the above characterization. The adjoint algebraic entropy of a linear transformation, its connection with the algebraic entropy of the adjoint map of the dual space and the dichotomy of its behavior are also illustrated.