Self-interacting polynomials

We introduce a class of dynamical systems of algebraic origin, consisting of self-interacting irreducible polynomials over a field. A polynomial f is made to act on a polynomial g by mapping the roots of g. This action identifies a new polynomial h, as the minimal polynomial of the displaced roots. By allowing several polynomials to act on one another, we obtain a self-interacting system with a rich dynamics, which affords a fresh viewpoint on some algebraic dynamical constructs. We identify the basic invariant sets, and study in some detail the case of quadratic polynomials. We perform some experiments on self-interacting polynomials over finite fields.