On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

Abstract

In this article we investigate the coprimeness properties of one and twodimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we shall focus on some higher power extensions of the Somos-4 equation and the (1-dimensional) discrete Toda equation. Our previous results are that all of the equations satisfy the irreducibility and the coprimeness properties if the r.h.s. is not factorizable. In this paper we shall prove that the coprimeness property still holds for all of these equations even if the r.h.s. is factorizable, although the irreducibility property is no longer satisfied.

Toda type equations over multi-dimensional lattices

“…which is equivalent to the coprimeness-preserving 2D discrete Toda equation (1). The coprimeness property of equation ( 7) is already proved in [17]. The statement is recast in the appendix B as theorem B.2.…”

“…When a = b = 1, equation (2) is equivalent to the coprimeness-preserving extensions to the two-dimensional Toda lattice equation (1). The degree growth of the iterates τ t,n,m of (1) is proved to be exponential unless k 1 = k 2 = l 1 = l 2 = 1 in our previous work [14,17]. Similarly, it is easy to prove that the degrees deg τ t,n of (2) grow exponentially with respect to t, unless (a, b) = (1, 1) and k 1 = k 2 = l 1 = l 2 = 1.…”

“…The statement is recast in the Appendix B as theorem B.2. The proof is found in [14,17]. Therefore v n = v m,n is coprime with every z m ′ ,n ′ , for the initial condition such that x, y depend only on m, n. Since z m = z m,n is not a unit, we have α m = 0.…”

An exercise in experimental mathematics: calculation of the algebraic entropy of a map

“…At this point it is interesting to notice that the recurrence relation ( 8) belongs the 'Somos-like' family (see [25,26,27] and https://faculty.uml.edu//jpropp/somos/history.txt ). Although A k+1 is given as a rational fraction in terms of the previous A's, the recurrence has the so called Laurent property [19,28]. Moreover, if one launches the recurrence with appropriate polynomial initial conditions, the values one obtains are, by construction, multivariate polynomials.…”

On the degree growth of iterated birational maps

“…Start from a generic point p 0 = [x, y, z, t] and examine the structure of the iterates of the map φ, calculated explicitly, keeping track of the product structure of the components, along the lines of [21], see also [22]:…”