English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $f$ be a polynomial over a global field $K$. For each $\alpha$ in $K$ and$N$ in $\mathbb{Z}_{\geq 0}$ denote by $K_N(f,\alpha)$ the arboreal field$K(f^{-N}(\alpha))$ and by $D_N(f,\alpha)$ its degree over $K$.  It is conjectured that $D_N(f,\alpha)$ should grow as a double exponentialfunction of $N$, unless $f$ is post-critically finite (PCF), in which casethere are examples like $D_N(x^2,\alpha) \leq 4^{N}$. There is evidenceconditionally on Vojta's conjecture. However, before the present work, nounconditional non-trivial lower bound was known for post-critically infinite$f$. In the case $f$ is PCF, no non-trivial lower bound was known, not evenunder Vojta's conjecture.  In this paper we give two simple methods that turn the finiteness of thecritical orbit into an exploitable feature, also in the post-criticallyinfinite case. First, assuming GRH for number fields, we establish for all PCFpolynomials $f$ of degree at least $2$ and all $\alpha$ outside of the criticalorbits of $f$, the existence of a positive constant $c(f,\alpha)$ such that$$D_N(f,\alpha) \geq \text{exp}(c(f,\alpha)N), $$ for all $N$, which is sharpup to, possibly, improve the constant $c(f,\alpha)$.  Second, we show unconditionally that if $f$ is post-critically infinite overany number field $K$ and unicritical, then, for each $\alpha$ in $K$, thereexists a positive constant $c(f,\alpha)$ such that $$D_N(f,\alpha) \geq\text{exp}(c(f,\alpha)N), $$ for all $N$. The main input here is to work moduloa suitably chosen prime and use a construction available for PCF unicriticalpolynomials with periodic critical orbit.

The size of arboreal images, I: exponential lower bounds for PCF and unicritical polynomials