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/

Zendy Plus

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

Zendy Tools

Zendy Open

A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime   $p$  , reduce its coefficients mod   $p$   and consider its action on the field   $ {{\mathbb{F}}}_p$  . The questions of whether and in what sense these families are random have been studied extensively, spurred in part by Pollard’s famous “rho” algorithm for integer factorization (the heuristic justification of which is the conjectural randomness of one such family). However, the cycle structure of these families  cannot  be random, since in any such family, the number of cycles of a fixed length in any dynamical system in that family is bounded. In this paper, we show that the cycle statistics of many of these families are  as random as possible . As a corollary, we show that most members of these families have many cycles, addressing a conjecture of Mans  et al .

The Cycle Structure of Unicritical Polynomials