Yann Bugeaud, Andrej Dujella
We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).
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In this work, we prove that the growth of the Artin conductor is at most, exponential in the degree of the character.
We prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values
We construct parametric families of (monic) reducible polynomials having two roots very close to each other.
While the separation (the minimal nonzero distance) between roots of a polynomial is a classical topic, its absolute counterpart (the minimal nonzero distance between their absolute values) does not seem to have been studied much. We present the general context and give tight bounds for the case of real roots.
An Elementary Problem in Galois Theory about the Roots of Irreducible Polynomials
本文研究了不可约多项式在单根生成域中的根数量问题，提出构造方法生成满足特定根数量关系的不可约多项式，并在有理数域上给出了具体应用与新问题。
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 …
On polynomials with roots modulo almost all primes
该研究定义了在几乎所有素数模下有根但无整数根的首一整系数多项式为例外多项式，分类了与不可约首一二次多项式乘积为例外多项式的不可约首一整系数多项式，并构造了指定因式形式的例外多项式。
We study root separation of reducible monic integer polynomials of odd degree. Let h(P) be the naive height, sep(P) the minimal distance between two distinct roots of an integer polynomial P(x) and sep(P)=h(P)^{-e(P)}. Let e_r*(d)=limsup_{deg(P)=d, h(P)-> +infty} e(P), where the limsup is taken over the reducible monic integer polynomials P(x) of degree d. We prove that e_r*(d) <= d-2. We also obtain a lower bound for e_r*(d) for d odd, which improves previously known lower bounds for e_r*(d) when d = 5, 7, 9.
Let f(x) be a primitive irreducible polynomial with integer coefficients of degree greater than one. In 1964, Hooley showed that the sequence of normalized roots u/n, where f(u) = 0(n), ordered in the obvious way, is uniformly distributed modulo one. It is the goal of this paper to show that if f(x) and g(x) are a pair of primitive irreducible polynomials of degree greater than one, not necessarily distinct, then the sequence (u/n,v/n), with f(u) = 0(n) and g(v) = 0(n), ordered in the obvious way, is uniformly distributed modulo one in the unit torus.
Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens Höchsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of a…
Let $\mathbb{F}_q$ denote the finite field of $q$ elements and $\mathbb{F}_{q^n}$ the degree $n$ extension of $\mathbb{F}_q$. A normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F} _q$ is a basis of the form $\{α,α^q,\dots,α^{q^{n-1}}\}$. An irreducible polynomial in $\mathbb{F} _q[x]$ is called an $N$-polynomial if its roots are linearly independent over $\mathbb{F} _q$. Let $p$ be the characteristic of $\mathbb{F} _q$. Pelis et al. showed that every monic irreducible polynomial with degree $n$ and nonzero trace is an $N$-polynomial provided that $n$ is either a power of $p$ or a prime differe…
Counting integer polynomials with several roots of maximal modulus
该研究对固定次数、高度有界且恰有指定数量最大模根的首一整系数多项式集合基数给出上下界，分别计数可约与不可约情形，得到全根等模的不可约多项式在奇偶次数下的数量估计。
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…
We give a separation bound for the complex roots of a trinomial $f \in \mathbb{Z}[X]$. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of $f$; in particular, it is polynomial in $\log (°f)$. It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of $f$ rather than the number of monomials) give separation bounds that are exponentially worse.As an algorithmic application, we show that the number of real roots of a trinomial $f$ can be comput…
Upper Bounds on Polynomial Root Separation
本文证明了复系数多项式根分离度以Mahler测度和次数表示的上界，并指出该界在常数因子意义下最优，同时探讨了实根情形下的常数改进。
Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le r_n<p$ and propose several conjectures on the distribution of an integer $\lceil \sum_{i\in S} r_i/p\rceil$ for a subset $S$ of $\{1,\dots,n\}$ when $p\to\infty$.
Based upon the general theory, developed by the author, on the parametrization of the irreducible representations of the hyper special compact groups corresponding to the regular adjoint orbit, supercuspidal representations of $SL_n(F)$ are explicitly constructed for which the formal degree conjecture and the root number conjecture are verified with respect to certain $L$-parameter defined, by means of Kaletha, that is, the local Langlands correspondence of tori and the Langlands-Schelstad procedure, by the data parametrizing the irreducible representations of the hyper special compact subgrou…
Absolute root separation
该研究针对整系数多项式的绝对根分离问题，改进了现有相关下界，并通过低次数实验发现当前界仍偏保守。
We present a structure theorem for the multiple non-cyclotomic irreducible factors appearing in the family of all univariate polynomials with a given set of coefficients and varying exponents. Roughly speaking, this result shows that the multiple non-cyclotomic irreducible factors of a sparse polynomial, are also sparse. To prove this, we give a variant of a theorem of Bombieri and Zannier on the intersection of a fixed subvariety of codimension 2 of the multiplicative group with all the torsion curves, with bounds having an explicit dependence on the height of the subvariety. We also use this…
p-adic root separation and the discriminant of integer polynomials
研究固定次数和有界高度的整数多项式的p-adic根分离问题，以及判别式被素数幂整除的计数问题，证明了存在大量具有几乎素数幂判别式的不可约多项式。
Consider a random polynomial $$ G_Q(x)=ξ_{Q,n}x^n+ξ_{Q,n-1}x^{n-1}+...+ξ_{Q,0} $$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\{-Q, ..., Q\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show that there exists a constant $C_n$, depending on $n$ only such that for all $Q\ge 2$ the distribution of $D(G_Q)$ can be approximated as follows $$ \sup_{-\infty\leq a\leq b\leq\infty}|\mathbb{P}(a\leq \frac{D(G_Q)}{Q^{2n-2}}\leq b)-\int_a^bφ_n(x)\, dx|\leq\frac{C_n}{\log Q}, $$ where $φ_n$ denotes the distribution function of the discriminant of a random polynomial …
Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\mathbb{F}_q$ of characteristic $p$ (equivalently, constructing the bigger field $\mathbb{F}_{q^{r^e}}$). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some new connections between these two problems and their variants. In 1897, Stickelberger proved that if a polynomial has an odd number of even degree factors, then its …
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