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1. Number of real roots

    1. Deterministic criteria for asymptotics of real roots


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      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


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          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)


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              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.


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                  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


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                      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).


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                          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) ?


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                              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.