scholarly journals Orbit counting in polarized dynamical systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Wade Hindes
Keyword(s):  
Dynamical Systems ◽  
Finitely Generated ◽  
Projective Varieties ◽  
Infinite Sets ◽  
Unicritical Polynomials

<p style='text-indent:20px;'>We extend recent orbit counts for finitely generated semigroups acting on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{P}^N $\end{document}</tex-math></inline-formula> to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.</p>

scholarly journals Gorensteinness and iteration of Cox rings for Fano type varieties

2022 ◽  
Author(s):  
Lukas Braun
Keyword(s):  
Class Group ◽  
Reductive Group ◽  
Finitely Generated ◽  
Projective Varieties ◽  
Cox Rings ◽  
Cox Ring

AbstractWe show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones X, iteration of Cox rings is finite with factorial master Cox ring. In particular, even if the class group has torsion, we can express such X as quotients of a factorial canonical quasicone by a solvable reductive group.

scholarly journals The Cycle Structure of Unicritical Polynomials

2018 ◽  
Vol 2020 (23) ◽  
pp. 9120-9147
Author(s):  
Andrew Bridy ◽  
Derek Garton
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Integer Factorization ◽  
Cycle Structure ◽  
Integer Coefficients ◽  
Fixed Length ◽  
Number Of Cycles ◽  
Unicritical Polynomials ◽  
Mod P

Abstract 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.

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