1. Number of real roots
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Deterministic criteria for asymptotics of real roots
Problem 1.1.
[Igor Wigman] For any $\epsilon>0$, give sufficient, explicit (efficiently checkable) conditions, holding with probability $\ge 1-\epsilon$ for a sequence of iid standard Gaussians, on a deterministic sequence $\{a_k\}_{0\le k\le n}$ of real numbers for the associated polynomial $p_n(z) = \sum_{0\le k\le n} a_k z^k$ to have $(1+O(\epsilon)) \frac2\pi \log n$ real roots. -
Almost sure convergence for Kac polynomials
Problem 1.2.
[Igor Pritsker] Let $N_n(\mathbb R)$ be the number of real zeros for the Kac polynomial of degree $n$ with iid standard normal coefficients. It is known that $N_n(\mathbb R) / \log n \to 2/\pi$ in probability.
a) Prove it converges almost surely.
b) Extend to non-Gaussian coefficients and study error terms. -
Sub-leading order for $\mathbb E N_n(\mathbb R)$
Problem 1.3.
[Yen Do] Let $N_n(\mathbb R)$ be the number of real zeros for a Kac polynomial with coefficients that are iid copies of a real variable $\xi$. Show $\mathbb E N_n(\mathbb R) = \frac2\pi \log n + C_\xi + o(1)$, where $C_\xi$ is a constant depending on the distribution of $\xi$. If possible, give a formula for $C_\xi$. This is known in the Gaussian case as well as for some discrete distributions. -
Problem 1.4.
[Frank Calegari] For the standard Gaussian Kac polynomial $P_n(z)$ with real coefficients, let $p_n(k)$ be the probability that the $k$ roots of largest modulus are all real.
i) Show $p(k):= \lim_{n\to \infty} p_n(k)$ exists.
ii) Estimate $p(1)$.
iii) Find asymptotics for $p(k)$ as $k\to \infty$. -
Variance and CLT for real roots of random orthogonal polynomials
Problem 1.5.
[Igor Pritsker] Let $p_k$ be the orthonormal polynomials for a measure $\nu$ on $\mathbb R$ that is absolutely continuous with respect to Lebesgue measure and whose support is a compact interval. Let $P_n(z) = \sum_{0\le k\le n} a_k p_k(z)$, with $a_k$ iid standard (real) Gaussians. As usual we write $N_n(\mathbb \R)$ for the number of real zeros of $P_n$.
i) Show $\frac1nVar N_n(\mathbb R) \to c$ for some constant $c$.
ii) Use this to establish a CLT for $N_n(\mathbb R)$. -
Problem 1.6.
[Van Vu (via Yen Do)] For various sequences of polynomials $P_n(z)$ with independent real coefficients we have $$0< \overline{\underline{\lim}}_{n\to \infty} \frac{ Var N_n(\mathbb R)}{ \mathbb E N_n(\mathbb R)} <\infty.$$ Give general conditions on sequences of coefficients (in particular on the sequence of their variances) for this to hold. -
Intermediate growth of $\mathbb E N_n(\mathbb R)$ ?
Problem 1.7.
[Oanh Nguyen] Fix $\epsilon\in (0,1/2)$. Find a bounded sequence $c_k\in \mathbb R$ such that the expected number of real roots of $\sum_{0\le k\le n} c_k \xi_k z^k$ is $n^{\epsilon+o(1)}$, where $\xi_k$ are iid standard Gaussians.
Cite this as: AimPL: Zeros of random polynomials, available at http://aimpl.org/randpolyzero.