Nicole R. Looper
We prove Odoni's conjecture in all prime degrees; namely, we prove that for every positive prime $p$, there exists a degree $p$ polynomial $φ\in\mathbb{Z}[x]$ with surjective arboreal Galois representation. We also show that Vojta's conjecture implies the existence of such a polynomial in many degrees $d$ which are not prime.
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Suppose $f \in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm{T}$ of $0$ under $f$. The resulting homomorphism $ϕ_f: \mathrm{Gal}_K \to \mathrm{Aut} \mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $ϕ_f$ is surjective. We show that this conjecture is false.
In 1985, Odoni showed that in characteristic $0$ the Galois group of the $n$-th iterate of the generic polynomial with degree $d$ is as large as possible. That is, he showed that this Galois group is the $n$-th wreath power of the symmetric group $S_d$. We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.
Characterizing monogenic trinomials $\boldsymbol{x^{12}+ax^6+b}$ according to their Galois groups
研究单基因三元多项式 $x^{12}+ax^6+b$ 的伽罗瓦群特征，给出不同伽罗瓦群下的具体描述，扩展了对幂组合四次和六次三元多项式单基因性的研究。
Monogenic even sextic trinomials and their Galois groups
本文研究了形如$f(x)=x^6+Ax^{2k}+B$的单基因六次三元多项式，探讨其伽罗瓦群结构及生成的六次域特性。
Let $K$ be a number field, and let $d\geq 2$. A conjecture of Odoni (stated more generally for characteristic zero Hilbertian fields $K$) posits that there is a monic polynomial $f\in K[x]$ of degree $d$, and a point $x_0\in K$, such that for every $n\geq 0$, the so-called arboreal Galois group $Gal(K(f^{-n}(x_0))/K)$ is an $n$-fold wreath product of the symmetric group $S_d$. In this paper, we prove Odoni's conjecture when $d$ is even and $K$ is an arbitrary number field, and also when both $d$ and $[K:Q]$ are odd.
Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups
本文研究整数系数六次三项式 $x^6+Ax^3+B$ 的单基因性，利用 Jakhar–Khanduja–Sangwan 定理按伽罗瓦群分类给出所有单基因情形的显式描述，并分析其生成的六次域何时不同。
Let~$E$ be a Hilbertian field of characteristic~$0$. R.W.K. Odoni conjectured that for every positive integer~$n$ there exists a polynomial~$f\in E[X]$ of degree~$n$ such that each iterate~$f^{\circ{k}}$ of~$f$ is irreducible and the Galois group of the splitting field of~$f^{\circ k}$ is isomorphic to the automorphism group of a regular,~$n$-branching tree of height~$k.$ We prove this conjecture when~$E$ is a number field.
We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.
Given a field $K$, a polynomial $f \in K[x]$, and a suitable element $t \in K$, the set of preimages of $t$ under the iterates $f^{\circ n}$ carries a natural structure of a $d$-ary tree. We study conditions under which the absolute Galois group of $K$ acts on the tree by the full group of automorphisms. When $K=\mathbb{Q}$ we exhibit examples of polynomials of every even degree with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of $K=F(t)$ and $f \in F[x]$ in which the corresponding Galois groups are the monodromy groups of the r…
Monogenic trinomials of the form $x^4+ax^3+d$ and their Galois groups
本文研究形如 $x^4+ax^3+d$ 的单生成整系数多项式的伽罗瓦群，给出仅依赖 $a$ 和 $d$ 的判别条件，并刻画 $G=D_4$ 与 $G=A_4$ 的情形，扩展了先前四次多项式的工作。
Iterates of prime producing polynomials and their Galois groups
本文研究有限域上多项式复合 $F(t,ϕ)$ 的素数生成性质及其迭代 Galois 群，证明在适当不可约特化下其可在大有限域中生成素数，并满足 Odoni 猜想、对应树状 Galois 表示满射。
Monogenic Cyclic Quartic Trinomials
本文证明了恰好存在三个形如 $x^4+bx^2+d$ 且 Galois 群为四阶循环群的单生成三项式。
In this paper, we determine mod $2$ Galois representations $\barρ_{ψ,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5ψX_0X_1X_2X_3X_4=0,\ ψ\in K$$ defined over a number field $K$ under the irreducibility condition of the quintic trinomial $f_ψ$ below. Applying this result, when $K=F$ is a totally real field, for some at most qaudratic totally real extension $M/F$, we prove that $\barρ_{ψ,2}|_{G_M}$ is associated to a Hilbert-Siegel modular H…
Iterates of polynomials over $\F_q(t)$ and their Galois groups
本文在有限域上的一元有理函数域上证明了 Odoni 猜想的类似形式，即存在多项式使其所有迭代不可约且伽罗瓦群为对称群的迭代圈积，并给出实例说明大多数多项式满足条件。
A Note on Abelian Monogenic Trinomials
本文确定了所有形如 $x^{2n}+ax^{n}+b$ 的阿贝尔单基因三项式，其中 $n,a,b\in{\mathbb Z}$，$n\ge 1$，$ab\ne 0$。
Monogenic cyclic trinomials of the form $x^4+cx+d$
本文证明了不存在形如 $x^4+cx+d$ 的单生成循环三项式，并结合先前研究确定了仅有的三个单生成循环四次三项式。
Fixed-point proportion of geometric iterated Galois groups
本文通过群论、遍历理论等方法，证明切比雪夫多项式是唯一具有正固定点比例的复多项式，解决了几何迭代伽罗瓦群的固定点比例问题。
On the monogenicity and Galois groups of $\boldsymbol{x^{2p}+ax^p+b^p}$
本文研究了三元式 $x^{2p}+ax^p+b^p$ 的单生成性及其伽罗瓦群特征，扩展了相关领域的研究成果。
The inverse Galois problem of iterated Galois groups and their fixed-point proportion
本文引入几乎混合群概念，研究迭代伽罗瓦群的逆伽罗瓦问题，证明非动力学拉回的有理函数几何迭代伽罗瓦群不动点比例为零，解决相关开放问题。
For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set of primes $p$ such that all iterations of $f$ are irreducible modulo $p$ is of relative density zero. Furthermore, we give an explicit bound on the rate of the decay of the density of such primes in an interval $[1, Q]$ as $Q \to \infty$. For this class of polynomials this gives a more precise version of a recent result of A. Ferraguti (2018), which applies t…
Monogenic Even Cyclic Sextic Polynomials
本文证明不存在单基因偶循环六次二项式或三项式，并在三类无限循环六次四项式中确定恰好四个单基因且分裂域不同的例子，同时说明非循环伽罗瓦群情形差异显著。
The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(α)}(z)$ associated with the dynamical zeta functions $ζ_{house(α)}(z)$ of the Rényi--Parry arithmetical dynamical systems, for $α$ an algebraic integer $α$ of house "$house(α)$" greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{house(α)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(α)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(house(α))$ of the Mahler…
The Conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{\houseα}(z)$ associated with the dynamical zeta functions $ζ_{\houseα}(z)$ of the Rényi--Parry arithmetical dynamical systems ($β$-shift), for $α$ a reciprocal algebraic integer of house $\houseα$ greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{\houseα}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $\houseα$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(\houseα)$ of …
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