Degree Sequences of Rational Maps Need Not Be Piecewise Linear Recurrent

Benedetto, Ingram, Jones, Manes, Silverman and Tucker asked whether, for every dominant rational map \(f\) of \(\mathbb{P}^N\), there are finitely many linear recursions with constant coefficients \(L^{(1)},\dots,L^{(r)}\) and an index function \(\iota\colon\mathbb{N}\to\{1,\dots,r\}\) such that \(\deg(f^n)=L^{(\iota(n))}_n\) for all \(n\ge1\) (Bull. Amer. Math. Soc. 56 (2019); Question 14.6 of arXiv:1806.04980). We read the question, as its notation suggests, as asking for finitely many fixed sequences, each satisfying a linear recursion with constant coefficients, with coefficients and initial values in any field of characteristic zero, and an arbitrary index function; the different reading in which the recursions are applied to the degree sequence itself, with coefficients that change with \(n\), is not addressed. Under our reading the answer is no. We show that if a sequence of positive integers \(a_n\) with \(a_n^{1/n}\to\delta\) agrees on the fibres of an index function with finitely many such sequences, then \(\delta=|\theta|\) for an algebraic number \(\theta\). The proof specializes the data to algebraic numbers and then uses the Skolem–Mahler–Lech theorem and Evertse's lower bound for sums of \(S\)-units, a consequence of the subspace theorem, in the form used by Bell, Diller and Jonsson. Bell, Diller and Jonsson (Acta Math. 2020) proved that an explicit dominant rational map of \(\mathbb{P}^2\), defined over \(\mathbb{Z}\), has a transcendental dynamical degree; hence its degree sequence has no such description. For \(N\ge3\) the same follows from the birational examples of Bell, Diller, Jonsson and Krieger. The deduction is short, and its novelty is modest. We also show that degree sequences of monomial maps are of the form in the question, which explains why the example of Hasselblatt and Propp mentioned in the source does not answer it. This is an unrefereed note.