Root separation for reducible integer polynomials

计数具有多个最大模根的整系数多项式

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。

$p$-进商集：具有可约特征多项式的线性递推序列

$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials

本文针对特征多项式在有理数域上可约的一类线性递推序列，解答了Sanna提出的判断其$p$-进商集在$p$-进数域中是否稠密的公开问题，补充了此前不可约情形下的相关研究结论。

On Hensel's roots and a factorization formula in Z[[x]]

Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. All of our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.

$\widetilde{SL_2(F)}$ 的可约幺正主级数表示的 Whittaker 空间

Whittaker spaces for reducible unitary principal series representations of $\widetilde{SL_2(F)}$

本文研究了 Kubota 构造的 $\widetilde{SL_2(F)}$ 的可约幺正真主级数表示，计算了其两个不可约直和项的 Whittaker 泛函空间维数，探讨了特征标修改的影响，并应用于确定扭曲 Kazhdan-Patterson 覆盖的作用，适用于分歧与非分歧表示。

多项式根分离的上界

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。

Some remarks on the non-real roots of polynomials

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $β_1 < \dots < β_m$ the distinct real roots of the discriminant $Δ_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $γ$ be the number of real roots of $g(x)=\sum_{k=0}^s t_{s-k} x^{s-k}$. For any $ξ> | β_m |$, if $n-s$ is odd then the number of real roots of $f(ξ, x)$ is $γ+1$, and if $n-s$ is even then the number of real roots of $f(ξ, x)$ is $γ$, $γ+2$ if $t_s>0$ or $t_s < 0$ respectively. A special case of the above result is constructing a family of totally complex polynomials which are reducible over $\m…

Root Separation for Trinomials

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…

绝对根分离

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。

p-adic根分离与整数多项式的判别式

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。

相交多项式与除数剖面的通用分离

Intersective Polynomials and Universal Separation of Divosor Profiles

该研究针对任意次数的互素多项式对及所有次数不超过2的多项式对，分类了除数剖面的通用分离问题，得出互素多项式对通用无界分离的充要条件，并解决了次数不超过2的多项式对的有限p进边界问题，证明结合了根进展的一致殆素数值与自适应局部路由论证。

Discriminant and root separation of integral polynomials

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 …

关于具有固定数域根的单项整数多项式自然密度的研究

On the Natural Density of Monic Integer Polynomials with Roots in a Fixed Number Field

本文研究了具有固定数域根的单项整数多项式族的统计分布和渐进行为，通过结合马赫勒测度、狄利克雷单位定理和迪克辛策数的残差分析，证明了密度趋于零的速度依赖于多项式次数，并建立了几何数论原理的显式组合界限，用于计数可约和不可约多项式。

关于本原二次多项式的Gow–McGuire猜想

Primitive Quadratic Polynomials in Additive Coset Families

该研究针对奇素数幂\textit{q}>204931的情况，结合特征和估计、改进素数筛法及精确有限计算，证明了Gow–McGuire关于本原二次多项式的猜想3，同时也证明了该范围内的猜想1和2。

Non-isogenous elliptic curves and hyperelliptic jacobians

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other reducible. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous ove…

希尔伯特定理90、周期性和Artin-Schreier多项式的根

Hilbert's Theorem 90, periodicity, and roots of Artin-Schreier polynomials

本文研究了特征为p>0的域扩张中希尔伯特定理90的周期性及Artin-Schreier多项式根的结构。