Silverman’s polynomial growth conjecture beyond monomial maps

Establish whether for every birational self-map ϕ of C^N with dynamical degree λ_ϕ, the normalized degree deg(ϕ^n)/λ_ϕ^n grows at most polynomially in n with degree at most N. Determine the validity of this conjecture for general (non-monomial) birational maps and dimensions beyond the already verified cases (monomial maps for all N and the integrable N = 2 case).

Background

The paper surveys known results on degree growth and singularity structures of birational mappings. While second-order cases are well classified, higher-order cases lack a general classification. In this context, a conjecture by Silverman proposes a universal polynomial upper bound (of degree at most N) for the normalized degree growth of iterates of birational maps on CN.

The authors note that Silverman’s conjecture has been established for monomial mappings in general dimension and for integrable mappings when N = 2, but they explicitly state that it remains open in all other cases. Resolving this would significantly advance the understanding of degree growth in higher-dimensional birational dynamics.

References

There is however an intriguing conjecture formulated by Silverman in [25], which can be roughly summarized as stating that for a birational mapping ϕ on CN with dynamical degree λϕ, the quantity deg(ϕn)/λϕn grows at most polynomially with n, for a polynomial with degree at most N. This conjecture has been shown to hold for monomial mappings for general N (and holds for integrable mappings, i.e. mappings for which λ = 1, when N = 2), but as far as the authors are aware of it is still an open problem in all other cases.

— Singularities and growth of higher order discrete equations  (2403.14329 - Willox et al., 2024) in Section 2, Some exact results, pp. 50–51