Rainer Dietmann
We show that there are at most $O_{n,ε}(H^{n-2+\sqrt{2}+ε})$ monic integer polynomials of degree $n$ having height at most $H$ and Galois group different from the full symmetric group $S_n$, improving on the previous 1973 world record $O_{n}(H^{n-1/2}\log H)$.
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On monic abelian trace-one cubic polynomials
本文研究了具有伽罗瓦群 $C_3$ 的首一迹一三次整系数多项式，利用环面几何高度函数和二次分圆域 $\mathbb Q(\sqrt{-3})$ 的参数化方法，计算了有界高度下的渐近数量，并给出了固定系数 $a$ 时多项式数量的显式公式。
In this paper we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over $\mathbb{Q}$ satisfying $\max\{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by $O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava-Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves
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。
Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynomials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with i…
Let $G$ be a subgroup of the symmetric group $S_n$, and let $δ_G=|S_n/G|^{-1}$ where $|S_n/G|$ is the index of $G$ in $S_n$. Then there are at most $O_{n, ε}(H^{n-1+δ_G+ε})$ monic integer polynomials of degree $n$ having Galois group $G$ and height not exceeding $H$, so there are only `few' polynomials having `small' Galois group.
We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{Ω(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.
In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo power…
Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group over a number field typically have splitting field with Galois group isomorphic to its Weyl group, we construct an explicit monic integral polynomial of degree 240 whose splitting field has Galois group the Weyl group of the exceptional group of type E_8.
We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2 (\log H)^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2 \log H)$, $O(H^2 \log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.
Counting Polynomials via Galois Actions on Root Subsets
本文研究了高度不超过H的单项整系数多项式数量，其伽罗华群在根上的作用同构于给定的置换群(G,Ω)。针对多种群族获得了新的上界。
Given two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's u…
We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{\mathfrak{p}}$ having residue field with $q= p^f$ elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when $q> n^2+n$. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed $n$, almost all Galois groups are cyclic in the limit $q \to \infty$. In particular, we show that the Galois groups are cyclic with p…
When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group isomorphic to $S_n$. In particular, amongst such polynomials whose coefficients are bounded by $B$ in absolute value, asymptotically $(1+o(1))(2B+1)^n$ are irreducible and have Galois group $S_n$. When $G$ is a proper transitive subgroup of $S_n$, however, the asymptotic count of polynomials with Galois group $G$ has been determined only in very few cases. Here, we s…
How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in $[-H,H]$, is $O(H^{3.91})$. More generally, we show that if $n \ge 3$ and $n \notin \{ 7, 8, 10 \}$ then there are $O(H^{n-1.017})$ monic, irreducible polynomials of degree $n$ with integer coefficients in $[-H,H]$ and Galois group not containing $A_n$. Save for the alternating group and degrees $7,8,10$, this establishes a 1936 conjecture of van der Waerden.
Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x]$ be a sequence of monic polynomials such that for every $n\in \mathbb N$, the composition $f^{(n)}=f_1\circ f_2\circ\ldots\circ f_n$ is irreducible. In this paper we show that if the size of the Galois group of $f^{(n)}$ is large enough (in a precise sense) as a function of $n$, then the set of primes $\mathfrak p\subseteq\mathcal O_K$ such that every $f^{(n)}$ is irreducible modulo $\mathfrak p$ has density zero. Moreover, we prove that the subset of polynomial sequences …
Galois theory by calculator
本文提出一种仅需判断整数是否为平方数以及多项式是否有整数根的计算器算法，用于确定次数不超过五的不可约首一整系数多项式的伽罗瓦群，并给出 Desmos 实现及适用于手持计算器的简化版本。
A note on Galois groups of linearized polynomials
本文研究线性化多项式L(X)的Galois群，证明当q为素数幂时，L(X)/X-t的Galois群为GL_n(q)，解决了关于偶数q的猜想。
Let $P(X)\in\mathbb{Z}[X]$ be an irreducible, monic, quartic polynomial with cyclic or dihedral Galois group. We prove that there exists a constant $c_P>0$ such that for a positive proportion of integers $n$, $P(n)$ has a prime factor $\ge n^{1+c_P}$.
Probabilistic Galois Theory -- The Square Discriminant Case
本文在大盒模型下研究随机多项式伽罗瓦群为 $A_n$ 的概率，提出比 Bhargava 的 $O(L^{-1})$ 更强的猜想上界 $L^{-n/2+\epsilon}$，并证明了该概率及判别式为平方数的相关概率的强下界。
Let $ϕ$ be a quadratic, monic polynomial with coefficients in $\mathcal O_{F,D}[t]$, where $\mathcal O_{F,D}$ is a localization of a number ring $\mathcal O_F$. In this paper, we first prove that if $ϕ$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_ϕ$ with the following property: for all primes $\mathfrak p\subseteq\mathcal O_{F,D}$ such that the reduced polynomial $ϕ_\mathfrak p\in (\mathcal O_{F,D}/\mathfrak p)[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of $ϕ_{\mathfrak p}$ is bounded by $N_ϕ$. Using this result, we prove th…
We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $ζ(2)^{-1}$. It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute discrim…
Full Galois groups of polynomials with slowly growing coefficients
本文研究整数系数首一多项式在系数盒子 $[-L,L]^n$ 中随机选取时伽罗瓦群为全对称群的概率，证明当 $L$ 随次数 $n$ 缓慢趋于无穷时该性质渐近几乎必然成立，并将方法推广到更一般的独立系数情形。
Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem
研究回文多项式Galois群的结构，发现其在一定比例下无法达到最大可能群S₂≀Sₙ，扩展了Bhargava对van der Waerden猜想的解决。
Let $K$ be a number field and $f\in K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear recurrence sequence associated to $f$, allowing sometimes to characterize the prime ideals of $O_K$ modulo which $f$ completely splits. If $α$ is a root of $f$, this criterion therefore gives a characterization of the prime ideals of $O_K$ which split completely in $K(α)$. It does apply if the degree of $f$ is at least $4$ and the Galois group of $f$ is the symmetric…
Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q_p for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m>1 irreducible polynomials, then its Galois group must be "m-coverable", i.e. a union of conjugates of m proper subgroups, whose total intersection is trivial. We are thus led to a variant of the inverse Galois problem: given an m-coverable finite group G, find a Galois realization of G over the rationals Q by a polynom…
Monogenic Quartic Polynomials and Their Galois Groups
本文借助 Kappe–Warren 的四次 Galois 群分类，为除循环群外的各类可能 Galois 群构造生成互异四次域的无限单参数单生四次多项式族，并讨论未解决的循环情形。
Landau's theorem asserts that the asymptotic density of sums of two squares in the interval $1\leq n\leq x$ is $K/{\sqrt{\log x}}$, where $K$ is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals $|n-x|\leq x^ε$ for a fixed $ε$ and $x\to \infty$. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic $f_0\in \mathbb{F}_q[T]$ of degree $n$ and take $ε$ with $1>ε\geq \frac2n$. Then the asymptotic density of polynomials $f$ in the `interval' ${…
Let $f(x)$ be a monic polynomial in $\dZ[x]$ with no rational roots but with roots in $\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more irreducible polynomials, and that if $f(x)$ is a product of $m>1$ irreducible polynomials, then its Galois group must be a union of conjugates of $m$ proper subgroups. We prove that for any $m>1$, every finite solvable group which is a union of conjugates of $m$ proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group …
Monogenic Even Octic Polynomials and Their Galois Groups
本文针对已分类的偶次八次多项式${\mathcal F}(x)$与${\mathcal G}(x)$的每个Galois群，构造了具有该群的monogenic多项式无限族，或证明并穷举了至多有限个此类族。
Probabilistic Galois Theory in Function Fields
本文研究函数域上随机多项式的不可约性与伽罗瓦群，证明其不可约概率趋于 $1-\frac{1}{q^d}$，并在多项式 Chowla 猜想下证明伽罗瓦群等于对称群 $S_n$。
Galois groups of random integer polynomials and van der Waerden's Conjecture
本文证明了范·德·瓦尔登猜想对所有次数 n 成立，即伽罗瓦群非全对称群的随机整数多项式数量上界为 O(H^(n-1))。
A proof of van der Waerden's Conjecture on random Galois groups of polynomials
本文概述了范德瓦尔登猜想的一个证明，该猜想断言系数有界的首一整系数多项式中伽罗瓦群非全对称群的数量上界为 $O(H^{n-1})$，从而将已知结果从 $n\leq 4$ 推广到所有次数。
On the Squarefree Values of Degree-$2q$ Polynomials
本研究证明满足特定伽罗瓦条件的2q次首一不可约整系数多项式有无穷多整数n使其值为无平方因子数，并据此构造了单位形状收敛到六边形格的非伽罗瓦全实三次域无穷子族。
For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we may also force the discriminant to be coprime to any given integer. The number of fields produced with discriminant in the range [-N, N] is at least c N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic numbe…
Kummer-faithful fields with finitely generated absolute Galois group
本文证明了在有限生成的绝对伽罗华群条件下，几乎所有数字段都是Kummer-忠实的，并展示了其在半阿贝尔簇Mordell-Weil群中的应用。
The monogenicity and Galois groups of certain reciprocal quintinomials
该研究拓展了互反五项式${\mathcal F}_{n,A,B}(x)$的单项性研究，解决了$A\equiv B\equiv 1\pmod{4}$这一此前未处理的特定情形，并确定了特殊情形下该多项式在有理数域上的伽罗瓦群。
We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z…
Abelian dynamical Galois groups for unicritical polynomials
本文研究单临界多项式的阿贝尔动力伽罗瓦群，将 Andrews–Petsche 猜想约化到后临界有限情形，并在周期临界轨道及二次数域首一情形下完整证明该猜想，同时给出例外集之外基点个数有限的结论。
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。
Let $F$ be a field of prime characteristic $p$ and let $q$ be a power of $p$. We assume that $F$ contains the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that those powers of $x$ that occur as terms of $L$ with nonzero coefficient have exponent a power of $q$. If the exponent of the leading term of $L$ is $q^n$, we say that $L$ has $q$-degree $n$. We assume that the coefficient of the $x$ term of $L$ is nonzero. We investigate the Galois group $G$, say, of $L$ over $F$, under the assumption that $L(x)/x$ is irreducible …
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。
Monogenic Cyclic Polynomials in Recurrence Sequences
本文研究多项式递推序列中单生成循环多项式的出现问题，明确了单生成与循环多项式的定义，属于数论领域的相关研究。
The inverse problem for arboreal Galois representations of index two
本文研究二次多项式树状Galois表示的逆问题，刻画指数2像子群的临界轨道判据，证明$x^2+t$恰有五种可能像并在Vojta猜想下均无限出现，且五子群拓扑互不同构。
Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short, very fast PARI Programs (§§3.1, 3.2) testing if K (defined by an irreducible monic polynomial) is p-rational or not. For quadratic fields we verify some densities related to Cohen-Lenstra-Martinet ones and analyse Greenberg's conjecture on the existence of p-rational fields with Galois groups (Z/2Z) t needed for the construction of some Galois representations with open image. We give examples for p = 3, t = 5 and t = 6 (§§5.1, 5.2) and …
The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field $K_f/\mathbb{Q}$ is the full symmetric group, has order $|\mathscr{P}_{n,N}^0(\mathbb{Q})|=(2N)^n+O_n(N^{n-1})$ as $N\rightarrow+\infty$. The conjecture has been shown previously for cubic and quartics polynomials by van der Waerden, Chow and Dietmann. Subsequently, Bhargava proved it for $n\ge6$. In this paper, we generalize the result for polynomials with coefficient…
Galois groups of integer polynomials with prescribed coefficients
本文研究固定部分系数的整数多项式伽罗瓦群，证明当自由系数足够多时，非对称群例外计数为H^{s-1}阶，结合原始群筛与稀疏Hermite插值，并处理低次端点情形。
Irreducibility and Galois groups of random reciprocal polynomials of large degree
该研究针对随机首一互反多项式，证明其在广泛概率测度下以接近1的概率不可约，且伽罗瓦群为超八面体群或其两个指数2子群，拓展了标准多项式不可约性与伽罗瓦群的相关结果。
Universality theorems for zeros of random real polynomials with fixed coefficients
该研究证明了含固定系数的随机首一实多项式恰有m个实零点的概率渐近公式，拓展了全随机系数情形的普遍性结果，并推导了n次代数整数实伽罗瓦共轭元计数的数论推论。
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 $n,a,b\in{\mathbb Z}$，$n\ge 1$，$ab\ne 0$。
Arakelov class groups of random number fields
本文为数域族中的Arakelov类群建立概率模型以修正Cohen--Lenstra--Martinet启发式，基于Chinburg的Omega(3)猜想对定向Arakelov类群的Galois模结构施加严格限制，并构造出具有非交换Galois群的无限反例系列。
Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the group…
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。
We discuss rather systematically the principle, implicit in earlier works, that for a "random" element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic polynomial, computed using any faitfhful representation, has Galois group isomorphic to the Weyl group of the underlying algebraic group. Besides tools such as the large sieve, which we had already used, we introduce some probabilistic ideas (large deviation estimates for finite Markov chains) and the general case involves a more precise understanding of the way…
Local-global conjugacy questions for affine extensions
研究Galois群$G_n(f)$在二元根树自同构群$\operatorname{Aut}(T_n)$中的局部-全局共轭性，通过上同调论证将Goksel的结果推广到置换表示的仿射扩张，并给出Goksel主要猜想的一个反例。
A Criterion for the Normality of Polynomials over Finite Fields Based on Their Coefficients
本文证明了存在基于系数的多项式 $h_n$，当不可约多项式代入系数后不为零时即可判定其正规性，并显式计算了 $n\le5$ 的情况及特征为 $p$ 时的简化判定多项式 $h_{p,n}$，该准则同样适用于具有循环伽罗瓦群的任意域。
Monte-Carlo Irreducibility and Imprimitivity Detection of Polynomials over $\mathbb{Q}$
本文提出了一种高效的蒙特卡洛算法，用于检测有理数域上多项式的算术不可约性，并通过子域提取提供构造性代数证书。
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