Abstract
We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:\mathbb{A}^n_\mathbb{C} \to \mathbb{A}^n_\mathbb{C}$ given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of “clusters” from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism $\sigma:\mathbb{C} \to \mathbb{C}$ those algebraic varieties $X \subseteq \mathbb{A}^n_\mathbb{C}$ for which $\Phi(X) \subseteq X^\sigma$. As a special case, we show that if $f(x) \in \mathbb{C}[x]$ is a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and $X \subseteq \mathbb{A}^2_\mathbb{C}$ is an irreducible curve that is invariant under the action of $(x,y) \mapsto (f(x),f(y))$ and projects dominantly in both directions, then $X$ must be the graph of a polynomial that commutes with $f$ under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius.
We also show that in models of ACFA$_0$, a disintegrated set defined by $\sigma(x) = f(x)$ for a polynomial $f$ has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$.
-
[ABR]
E. Amerik, F. Bogomolov, and M. Rovinsky, "Remarks on endomorphisms and rational points," Compos. Math., vol. 147, iss. 6, pp. 1819-1842, 2011.
@article {ABR, MRKEY = {2862064},
AUTHOR = {Amerik, E. and Bogomolov, F. and Rovinsky, M.},
TITLE = {Remarks on endomorphisms and rational points},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {147},
YEAR = {2011},
NUMBER = {6},
PAGES = {1819--1842},
ISSN = {0010-437X},
MRCLASS = {37P55 (11G35 11S82 14G05 37P20)},
MRNUMBER = {2862064},
MRREVIEWER = {Joseph H. Silverman},
DOI = {10.1112/S0010437X11005537},
ZBLNUMBER = {1231.14014},
} -
[AC]
E. Amerik and F. Campana, "Fibrations méromorphes sur certaines variétés à fibré canonique trivial," Pure Appl. Math. Q., vol. 4, iss. 2, part 1, pp. 509-545, 2008.
@article {AC, MRKEY = {2400885},
AUTHOR = {Amerik, Ekaterina and Campana, Fr{é}d{é}ric},
TITLE = {Fibrations méromorphes sur certaines variétés à fibré canonique trivial},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {4},
YEAR = {2008},
NUMBER = {2, part 1},
PAGES = {509--545},
ISSN = {1558-8599},
MRCLASS = {14D06 (14E05 14J10 32J27)},
MRNUMBER = {2400885},
MRREVIEWER = {Philippe P. Eyssidieux},
ZBLNUMBER = {1143.14035},
DOI = {10.4310/PAMQ.2008.v4.n2.a9},
} -
[Chat]
Z. Chatzidakis, "Groups definable in ACFA," in Algebraic Model Theory, Dordrecht: Kluwer Acad. Publ., 1997, vol. 496, pp. 25-52.
@incollection {Chat, MRKEY = {1481438},
AUTHOR = {Chatzidakis, Zo{é}},
TITLE = {Groups definable in {ACFA}},
BOOKTITLE = {Algebraic Model Theory},
VENUE={{T}oronto, {ON},
1996},
SERIES = {NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.},
VOLUME = {496},
PAGES = {25--52},
PUBLISHER = {Kluwer Acad. Publ.},
ADDRESS = {Dordrecht},
YEAR = {1997},
MRCLASS = {03C60},
MRNUMBER = {1481438},
ZBLNUMBER = {0884.03038},
DOI = {10.1007/978-94-015-8923-9_2},
} -
[CH-ACFA1]
Z. Chatzidakis and E. Hrushovski, "Model theory of difference fields," Trans. Amer. Math. Soc., vol. 351, iss. 8, pp. 2997-3071, 1999.
@article {CH-ACFA1, MRKEY = {1652269},
AUTHOR = {Chatzidakis, Zo{é} and Hrushovski, Ehud},
TITLE = {Model theory of difference fields},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {351},
YEAR = {1999},
NUMBER = {8},
PAGES = {2997--3071},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {03C60 (03C45 08A35 12L12)},
MRNUMBER = {1652269},
MRREVIEWER = {Salma Kuhlmann},
DOI = {10.1090/S0002-9947-99-02498-8},
ZBLNUMBER = {0922.03054},
} -
[CH-AD1]
Z. Chatzidakis and E. Hrushovski, "Difference fields and descent in algebraic dynamics. I," J. Inst. Math. Jussieu, vol. 7, iss. 4, pp. 653-686, 2008.
@article {CH-AD1, MRKEY = {2469450},
AUTHOR = {Chatzidakis, Zo{é} and Hrushovski, Ehud},
TITLE = {Difference fields and descent in algebraic dynamics. {I}},
JOURNAL = {J. Inst. Math. Jussieu},
FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
VOLUME = {7},
YEAR = {2008},
NUMBER = {4},
PAGES = {653--686},
ISSN = {1474-7480},
MRCLASS = {03C60 (12H10 14G25)},
MRNUMBER = {2469450},
MRREVIEWER = {David A. Pierce},
DOI = {10.1017/S1474748008000273},
ZBLNUMBER = {1165.03014},
} -
[CHP]
Z. Chatzidakis, E. Hrushovski, and Y. Peterzil, "Model theory of difference fields. II. Periodic ideals and the trichotomy in all characteristics," Proc. London Math. Soc., vol. 85, iss. 2, pp. 257-311, 2002.
@article {CHP, MRKEY = {1912052},
AUTHOR = {Chatzidakis, Zo{é} and Hrushovski, Ehud and Peterzil, Ya'acov},
TITLE = {Model theory of difference fields. {II}. {P}eriodic ideals and the trichotomy in all characteristics},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {85},
YEAR = {2002},
NUMBER = {2},
PAGES = {257--311},
ISSN = {0024-6115},
CODEN = {PLMTAL},
MRCLASS = {03C60 (03C45 08A35 12H05)},
MRNUMBER = {1912052},
MRREVIEWER = {Carlo Toffalori},
DOI = {10.1112/S0024611502013576},
ZBLNUMBER = {1025.03026},
} -
[GhTuZh]
D. Ghioca, T. J. Tucker, and S. Zhang, "Towards a dynamical Manin-Mumford conjecture," Int. Math. Res. Not., vol. 2011, iss. 22, pp. 5109-5122, 2011.
@article {GhTuZh, MRKEY = {2854724},
AUTHOR = {Ghioca, Dragos and Tucker, Thomas J. and Zhang, Shouwu},
TITLE = {Towards a dynamical {M}anin-{M}umford conjecture},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices. IMRN},
YEAR = {2011},
NUMBER = {22},
PAGES = {5109--5122},
ISSN = {1073-7928},
MRCLASS = {37P15 (11G10 37P55)},
MRNUMBER = {2854724},
MRREVIEWER = {Joseph H. Silverman},
DOI = {10.1093/imrn/rnq283},
VOLUME = {2011},
ZBLNUMBER = {1267.37110},
} -
[Hindry]
M. Hindry, "Autour d’une conjecture de Serge Lang," Invent. Math., vol. 94, iss. 3, pp. 575-603, 1988.
@article {Hindry, MRKEY = {0969244},
AUTHOR = {Hindry, Marc},
TITLE = {Autour d'une conjecture de {S}erge {L}ang},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {94},
YEAR = {1988},
NUMBER = {3},
PAGES = {575--603},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11G10 (11J89 14K15)},
MRNUMBER = {0969244},
MRREVIEWER = {Michel Laurent},
DOI = {10.1007/BF01394276},
ZBLNUMBER = {0638.14026},
} -
[Hr-MM]
E. Hrushovski, "The Manin-Mumford conjecture and the model theory of difference fields," Ann. Pure Appl. Logic, vol. 112, iss. 1, pp. 43-115, 2001.
@article {Hr-MM, MRKEY = {1854232},
AUTHOR = {Hrushovski, Ehud},
TITLE = {The {M}anin-{M}umford conjecture and the model theory of difference fields},
JOURNAL = {Ann. Pure Appl. Logic},
FJOURNAL = {Annals of Pure and Applied Logic},
VOLUME = {112},
YEAR = {2001},
NUMBER = {1},
PAGES = {43--115},
ISSN = {0168-0072},
CODEN = {APALD7},
MRCLASS = {03C60 (11G10 11U09 12L12)},
MRNUMBER = {1854232},
MRREVIEWER = {C. Toffalori},
DOI = {10.1016/S0168-0072(01)00096-3},
ZBLNUMBER = {0987.03036},
} -
[HI]
E. Hrushovski and M. Itai, "On model complete differential fields," Trans. Amer. Math. Soc., vol. 355, iss. 11, pp. 4267-4296, 2003.
@article {HI, MRKEY = {1990753},
AUTHOR = {Hrushovski, E. and Itai, M.},
TITLE = {On model complete differential fields},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {355},
YEAR = {2003},
NUMBER = {11},
PAGES = {4267--4296},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {03C60 (12H05)},
MRNUMBER = {1990753},
MRREVIEWER = {Wai Yan Pong},
DOI = {10.1090/S0002-9947-03-03264-1},
ZBLNUMBER = {1021.03024},
} -
@book {Med, MRKEY = {2710587},
AUTHOR = {Medvedev, Alice},
TITLE = {Minimal sets in {ACFA}},
NOTE = {Thesis (Ph.D.)--University of California, Berkeley},
PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
YEAR = {2007},
PAGES = {96},
ISBN = {978-0549-17089-1},
MRCLASS = {Thesis},
MRNUMBER = {2710587},
URL = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3275517},
ZBLNUMBER = {1235.03069},
} -
[selfarXiv] A. Medvedev and T. Scanlon, Polynomial dynamics, 2009.
@misc{selfarXiv,
author={Medvedev, Alice and Scanlon, T.},
TITLE={Polynomial dynamics},
ARXIV={0901.2352v2},
NOTE={68 pages},
YEAR={2009},
} -
[PR]
R. Pink and D. Roessler, "On $\psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture," J. Algebraic Geom., vol. 13, iss. 4, pp. 771-798, 2004.
@article {PR, MRKEY = {2073195},
AUTHOR = {Pink, Richard and Roessler, Damian},
TITLE = {On {$\psi$}-invariant subvarieties of semiabelian varieties and the {M}anin-{M}umford conjecture},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {13},
YEAR = {2004},
NUMBER = {4},
PAGES = {771--798},
ISSN = {1056-3911},
MRCLASS = {14K05},
MRNUMBER = {2073195},
MRREVIEWER = {Federica Galluzzi},
DOI = {10.1090/S1056-3911-04-00368-6},
ZBLNUMBER = {1072.14054},
} -
[Ritt-iteration]
J. F. Ritt, "On the iteration of rational functions," Trans. Amer. Math. Soc., vol. 21, iss. 3, pp. 348-356, 1920.
@article {Ritt-iteration, MRKEY = {1501149},
AUTHOR = {Ritt, J. F.},
TITLE = {On the iteration of rational functions},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {21},
YEAR = {1920},
NUMBER = {3},
PAGES = {348--356},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {30D05 (37F10)},
MRNUMBER = {1501149},
DOI = {10.2307/1988936},
JFMNUMBER = {47.0312.01},
} -
[Ritt]
J. F. Ritt, "Prime and composite polynomials," Trans. Amer. Math. Soc., vol. 23, iss. 1, pp. 51-66, 1922.
@article {Ritt, MRKEY = {1501189},
AUTHOR = {Ritt, J. F.},
TITLE = {Prime and composite polynomials},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {23},
YEAR = {1922},
NUMBER = {1},
PAGES = {51--66},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {13B25},
MRNUMBER = {1501189},
DOI = {10.2307/1988911},
JFMNUMBER = {48.0079.01},
} -
[Ritt-permutable]
J. F. Ritt, "Permutable rational functions," Trans. Amer. Math. Soc., vol. 25, iss. 3, pp. 399-448, 1923.
@article {Ritt-permutable, MRKEY = {1501252},
AUTHOR = {Ritt, J. F.},
TITLE = {Permutable rational functions},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {25},
YEAR = {1923},
NUMBER = {3},
PAGES = {399--448},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {30D05},
MRNUMBER = {1501252},
DOI = {10.2307/1989297},
JFMNUMBER = {49.0712.02},
} -
[ScICM]
T. Scanlon, "Analytic difference rings," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2007, pp. 71-92.
@incollection {ScICM, MRKEY = {2275589},
AUTHOR = {Scanlon, Thomas},
TITLE = {Analytic difference rings},
BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {II}},
VENUE={Madrid, August 22--30, 2006},
PAGES = {71--92},
PUBLISHER = {Eur. Math. Soc., Zürich},
YEAR = {2007},
MRCLASS = {03C10 (03C60 11G10 12H10 12J10 13K05)},
MRNUMBER = {2275589},
MRREVIEWER = {Cydara C. Ripoll},
ZBLNUMBER = {1104.03029},
DOI = {10.4171/022-2/4},
} -
[ScAO]
T. Scanlon, "Local André-Oort conjecture for the universal abelian variety," Invent. Math., vol. 163, iss. 1, pp. 191-211, 2006.
@article {ScAO, MRKEY = {2208421},
AUTHOR = {Scanlon, Thomas},
TITLE = {Local {A}ndré-{O}ort conjecture for the universal abelian variety},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {163},
YEAR = {2006},
NUMBER = {1},
PAGES = {191--211},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14K10 (11G18 14G35 14K15)},
MRNUMBER = {2208421},
DOI = {10.1007/s00222-005-0460-1},
ZBLNUMBER = {1086.14020},
} -
[Silverman]
J. H. Silverman, The Arithmetic of Dynamical Systems, New York: Springer-Verlag, 2007, vol. 241.
@book {Silverman, MRKEY = {2316407},
AUTHOR = {Silverman, Joseph H.},
TITLE = {The Arithmetic of Dynamical Systems},
SERIES = {Grad. Texts in Math.},
VOLUME = {241},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2007},
PAGES = {x+511},
ISBN = {978-0-387-69903-5},
MRCLASS = {11-02 (11-01 11G05 11G07 11G50 37-02 37F10)},
MRNUMBER = {2316407},
MRREVIEWER = {Thomas Ward},
DOI = {10.1007/978-0-387-69904-2},
ZBLNUMBER = {1130.37001},
} -
[Zh] S. Zhang, "Distributions in algebraic dynamics," in Surveys in Differential Geometry. Vol. X, Int. Press, Somerville, MA, 2006, vol. 10, pp. 381-430.
@incollection {Zh, MRKEY = {2408228},
AUTHOR = {Zhang, Shou-Wu},
TITLE = {Distributions in algebraic dynamics},
BOOKTITLE = {Surveys in Differential Geometry. {V}ol. {X}},
SERIES = {Surv. Differ. Geom.},
VOLUME = {10},
PAGES = {381--430},
PUBLISHER = {Int. Press, Somerville, MA},
YEAR = {2006},
MRCLASS = {32H50 (11G50 32J27 37F10)},
MRNUMBER = {2408228},
MRREVIEWER = {Matthew H. Baker},
ZBLNUMBER = {1207.37057},
} -
[MZ] M. Zieve and P. Müller, On Ritt’s polynomial decomposition theorems, 2008.
@misc{MZ,
author={Zieve, M. and Müller, P.},
TITLE={On {R}itt's polynomial decomposition theorems},
ARXIV={0807.3578v1},
NOTE={38 pages},
YEAR={2008},
}