Integrality and sharp upper bound for Silverman’s polynomial degree-growth exponent

Establish whether, for every dominant rational self-map f of projective space P^d, the quantity ell_f defined by the infimum of exponents controlling deg_{1,O(1)}(f^n)/(n^{ell} lambda_1(f)^n) exists as an integer satisfying 0 <= ell_f <= d.

Background

For a dominant rational self-map of projective space, Silverman’s exponent ell_f measures the smallest polynomial correction to the dominant exponential growth rate of the first-degree sequence. The paper proves a finite polynomial upper bound, namely ell_f <= 2(d-1), but explicitly notes that this does not establish the conjectured integrality or the sharper bound ell_f <= d.

The conjecture concerns the finer asymptotic behavior of degree growth beyond the dynamical degree. It remains relevant even after the paper rules out subexponential but superpolynomial growth, because the precise polynomial exponent is not determined by the paper’s estimates.

References

For a dominant rational self-map $f$ of $Pd$, Silverman Conjecture~2 conjectured that the quantity

\ell_f \coloneqq \inf\biggl{\ell\geq0:\sup_{n\geq1} \frac{\deg_{1,O(1)}(fn)}{n\ell \lambda_1(f)n}<\infty\biggr}

exists and is an integer with $0\leq \ell_f\leq d$.

— Degree growth of rational and meromorphic self-maps  (2610.18094 - Hu, 13 Oct 2026) in Remark following Corollary no-intermediate-growth, Introduction

\item Integrality. If the limit exists, is $\mu_i(f)\inZ_{\geq0}$?

— Degree growth of rational and meromorphic self-maps  (2610.18094 - Hu, 13 Oct 2026) in Question polynomial-dyn-deg, Section Further consequences and sharpness