Powers in Orbits of Rational Functions: Cases of an Arithmetic Dynamical Mordell–Lang Conjecture
Let $K$ be a finitely generated field of characteristic zero. For fixed $m\geqslant 2$ , we study the rational functions $\unicode[STIX]{x1D719}$ defined over $K$ that have a $K$ -orbit containing infinitely many distinct $m$ -th powers. For $m\geqslant 5$ we show that the only such functions are th...
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| Vydáno v: | Canadian journal of mathematics Ročník 71; číslo 4; s. 773 - 817 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Canada
Canadian Mathematical Society
01.08.2019
Cambridge University Press |
| Témata: | |
| ISSN: | 0008-414X, 1496-4279 |
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| Abstract | Let
$K$
be a finitely generated field of characteristic zero. For fixed
$m\geqslant 2$
, we study the rational functions
$\unicode[STIX]{x1D719}$
defined over
$K$
that have a
$K$
-orbit containing infinitely many distinct
$m$
-th powers. For
$m\geqslant 5$
we show that the only such functions are those of the form
$cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$
with
$\unicode[STIX]{x1D713}\in K(x)$
, and for
$m\leqslant 4$
we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set
$\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$
is a union of finitely many arithmetic progressions, where
$\unicode[STIX]{x1D719}^{n}$
denotes the
$n$
-th iterate of
$\unicode[STIX]{x1D719}$
and
$\unicode[STIX]{x1D706}\in K(x)$
is any map Möbius-conjugate over
$K$
to
$x^{m}$
. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves
$y^{m}=\unicode[STIX]{x1D719}^{n}(x)$
. We describe all
$\unicode[STIX]{x1D719}$
for which these curves have an irreducible component of genus at most 1, and show that such
$\unicode[STIX]{x1D719}$
must have two distinct iterates that are equal in
$K(x)^{\ast }/K(x)^{\ast m}$
. |
|---|---|
| AbstractList | Let
$K$
be a finitely generated field of characteristic zero. For fixed
$m\geqslant 2$
, we study the rational functions
$\unicode[STIX]{x1D719}$
defined over
$K$
that have a
$K$
-orbit containing infinitely many distinct
$m$
-th powers. For
$m\geqslant 5$
we show that the only such functions are those of the form
$cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$
with
$\unicode[STIX]{x1D713}\in K(x)$
, and for
$m\leqslant 4$
we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before.
With additional analysis, we show that the index set
$\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$
is a union of finitely many arithmetic progressions, where
$\unicode[STIX]{x1D719}^{n}$
denotes the
$n$
-th iterate of
$\unicode[STIX]{x1D719}$
and
$\unicode[STIX]{x1D706}\in K(x)$
is any map Möbius-conjugate over
$K$
to
$x^{m}$
. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves
$y^{m}=\unicode[STIX]{x1D719}^{n}(x)$
. We describe all
$\unicode[STIX]{x1D719}$
for which these curves have an irreducible component of genus at most 1, and show that such
$\unicode[STIX]{x1D719}$
must have two distinct iterates that are equal in
$K(x)^{\ast }/K(x)^{\ast m}$
. Let $K$ be a finitely generated field of characteristic zero. For fixed $m\geqslant 2$ , we study the rational functions $\unicode[STIX]{x1D719}$ defined over $K$ that have a $K$ -orbit containing infinitely many distinct $m$ -th powers. For $m\geqslant 5$ we show that the only such functions are those of the form $cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$ with $\unicode[STIX]{x1D713}\in K(x)$ , and for $m\leqslant 4$ we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set $\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$ is a union of finitely many arithmetic progressions, where $\unicode[STIX]{x1D719}^{n}$ denotes the $n$ -th iterate of $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D706}\in K(x)$ is any map Möbius-conjugate over $K$ to $x^{m}$ . When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^{m}=\unicode[STIX]{x1D719}^{n}(x)$ . We describe all $\unicode[STIX]{x1D719}$ for which these curves have an irreducible component of genus at most 1, and show that such $\unicode[STIX]{x1D719}$ must have two distinct iterates that are equal in $K(x)^{\ast }/K(x)^{\ast m}$ . Let \(K\) be a finitely generated field of characteristic zero. For fixed \(m\geqslant 2\), we study the rational functions \(\unicode[STIX]{x1D719}\) defined over \(K\) that have a \(K\)-orbit containing infinitely many distinct \(m\)-th powers. For \(m\geqslant 5\) we show that the only such functions are those of the form \(cx^{j}(\unicode[STIX]{x1D713}(x))^{m}\) with \(\unicode[STIX]{x1D713}\in K(x)\), and for \(m\leqslant 4\) we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before.With additional analysis, we show that the index set \(\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}\) is a union of finitely many arithmetic progressions, where \(\unicode[STIX]{x1D719}^{n}\) denotes the \(n\)-th iterate of \(\unicode[STIX]{x1D719}\) and \(\unicode[STIX]{x1D706}\in K(x)\) is any map Möbius-conjugate over \(K\) to \(x^{m}\). When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves \(y^{m}=\unicode[STIX]{x1D719}^{n}(x)\). We describe all \(\unicode[STIX]{x1D719}\) for which these curves have an irreducible component of genus at most 1, and show that such \(\unicode[STIX]{x1D719}\) must have two distinct iterates that are equal in \(K(x)^{\ast }/K(x)^{\ast m}\). |
| Author | Spear, Jacob Jones, Rafe Cahn, Jordan |
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| References | Karpilovsky (S0008414X1800055X_r13) 1989 S0008414X1800055X_r11 S0008414X1800055X_r22 S0008414X1800055X_r12 S0008414X1800055X_r20 S0008414X1800055X_r21 S0008414X1800055X_r10 Stichtenoth (S0008414X1800055X_r23) 2009 S0008414X1800055X_r19 S0008414X1800055X_r17 S0008414X1800055X_r18 S0008414X1800055X_r9 S0008414X1800055X_r15 S0008414X1800055X_r16 S0008414X1800055X_r8 S0008414X1800055X_r7 S0008414X1800055X_r14 S0008414X1800055X_r6 S0008414X1800055X_r5 S0008414X1800055X_r4 S0008414X1800055X_r3 S0008414X1800055X_r2 S0008414X1800055X_r1 |
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| Snippet | Let
$K$
be a finitely generated field of characteristic zero. For fixed
$m\geqslant 2$
, we study the rational functions
$\unicode[STIX]{x1D719}$
defined over... Let \(K\) be a finitely generated field of characteristic zero. For fixed \(m\geqslant 2\), we study the rational functions \(\unicode[STIX]{x1D719}\) defined... |
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| Title | Powers in Orbits of Rational Functions: Cases of an Arithmetic Dynamical Mordell–Lang Conjecture |
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