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Eventually stable rational functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2017 (English)In: International Journal of Number Theory, ISSN 1793-0421, Vol. 13, no 9, p. 2299-2318Article in journal (Refereed) Published
Abstract [en]

For a field K, rational function phi is an element of K(z) of degree at least two, and alpha is an element of P-1(K), we study the polynomials in K[z] whose roots are given by the solutions in K to phi(n)(z) = a, where fn denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (phi, alpha) is called eventually stable over K. We conjecture that (phi, alpha) is eventually stable over K when K is any global field and a is any point not periodic under f (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when K has a discrete valuation for which (1) f has good reduction and (2) facts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of S-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.

Place, publisher, year, edition, pages
WORLD SCIENTIFIC PUBL CO PTE LTD , 2017. Vol. 13, no 9, p. 2299-2318
Keywords [en]
Arithmetic dynamical systems, irreducibility of polynomials, S-integer points in dynamics
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-215785DOI: 10.1142/S1793042117501263ISI: 000411514700007Scopus ID: 2-s2.0-85019084989OAI: oai:DiVA.org:kth-215785DiVA, id: diva2:1150281
Note

QC 20171018

Available from: 2017-10-18 Created: 2017-10-18 Last updated: 2017-10-18Bibliographically approved

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  • apa
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