The expected number of zeros of a random system of $p$-adic polynomials

The Expected Number of Roots over The Field of p-adic Numbers

We study the roots of a random polynomial over the field of $p$-adic numbers. For a random monic polynomial with i.i.d. coefficients in $\mathbb{Z}_p$, we obtain an estimate for the expected number of roots of this polynomial. In particular, if the coefficients take the values $\pm1$ with equal probability, the expected number of $p$-adic roots converges to $\left(p-1\right)/\left(p+1\right)$ as the degree of the polynomial tends to $\infty$.

A $p$-adic Descartes solver: the Strassman solver

Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this paper, we translate this method into the $p$-adic worlds. We show how the $p$-adic analog of Descartes' rule of signs, Strassman's theorem, leads to an algorithm to isolate the roots of a square-free $p$-adic polynomial. Moreover, we show that this algorithm runs in $\mathcal{O}(d^2\log^3d)$-time for a random $p$-adic polynomial of degree $d$. To perform this analysis, we introduce the…

$p$-adic 随机矩阵在代数扩张中的特征值分布，以及 $p$-adic 随机多项式的类比

Eigenvalue Distribution of $p$-adic Random Matrices Among Algebraic Extensions, with an Analogue for $p$-adic Random Polynomials

研究 $p$-adic 随机矩阵在 $p$-adic 有理数域代数扩张中的特征值分布，通过 $p$-adic 类比于实数 Ginibre 集团的特征值计数结果，但具有不同的分布行为，同时证明最大无分支扩张 $\mathbb{Q}_p^{\mathrm{un}}$ 能捕捉几乎所有特征值，且超出该扩张的特征值数量有有限正极限。

Hall-Littlewood polynomials, boundaries, and $p$-adic random matrices

We prove that the boundary of the Hall-Littlewood $t$-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin and Cuenca on boundaries of related deformed Gelfand-Tsetlin graphs. In the special case when $1/t$ is a prime $p$ we use this to recover results of Bufetov-Qiu and Assiotis on infinite $p$-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions. Our methods rely on explicit formulas for certain skew Hall-Littlewood polynomials. As a separate corollary to these, we obtain …

基于结式分布方法的随机p进多项式与随机矩阵的普适性

The Resultant Distribution Method: Universality for $p$-adic Random Matrices and Polynomials

基于结式分布方法，证明了随机p进多项式零点及随机p进矩阵特征多项式的普适性，扩展了Haar模型公式，揭示了相关模型的鲁棒性。