Integrability of Difference Equations Through Algebraic Entropy and Generalized Symmetries

Coalgebra symmetry for discrete systems

“…A generating function is a predictive tool which can be used to test the successive members of a finite sequence. It follows that the algebraic entropy is given by the logarithm of the smallest pole of the generating function, see [25,27]. A birational map (or its avatar difference equation) will then be integrable if all the poles of the generating function lie on the unit circle.…”

Bi-rational maps in four dimensions with two invariants

“…A generating function is a predictive tool which can be used to test the successive members of a finite sequence. It follows that the algebraic entropy is given by the logarithm of the smallest pole of the generating function, see [18,19].…”

“…The parameter ν is shared with the main map (P.v). The map given by (Q.v) has the following degrees of iterates: 3,9,19,33,51,73,99,129,163 . .…”

Algebraic Entropy of a Class of Five-Point Differential-Difference Equations

“…called the algebraic entropy of the map ϕ When no confusion is possible about the map ϕ we will usually omit the subscript ϕ in (17). Algebraic entropy for bi-rational maps has the following properties [7,25,26]:…”

“…To practically compute the algebraic entropy we introduce some technical methods to reduce the computational complexity [26,27]. First, we fix the desired number of iterations to be some fixed N ∈ N. Following remark 3 this means that we need only finitely many initial conditions given by (15).…”