Are the largest roots of a random polynomial real?

Take a polynomial of degree $n$ whose coefficients are independent standard Gaussians. Most of its roots crowd near the unit circle and only about $\frac{2}{\pi}\log n$ are real. What is the chance that the root of largest absolute value is real, or that the $k$ largest are all real? Calegari asked whether these probabilities converge as $n$ grows, and what the limits are.

Problem

Let $\xi_0, \xi_1, \dots$ be independent standard real Gaussian random variables and let

$$P_n(z) = \sum_{j=0}^{n} \xi_j z^j$$

be the Gaussian Kac polynomial of degree $n$. Order its roots by decreasing absolute value, and let $p_n(k)$ be the probability that the $k$ roots of largest absolute value are all real.

1. Show that $p(k) = \lim_{n \to \infty} p_n(k)$ exists.
2. Estimate $p(1)$.
3. Find the asymptotic behaviour of $p(k)$ as $k \to \infty$.

Background

The question was posed by Frank Calegari at the AIM workshop on zeros of random polynomials. Kac (1943) showed that the expected number of real roots of $P_n$ is asymptotic to $\frac{2}{\pi}\log n$, so real roots are rare, while the roots as a whole concentrate near the unit circle. The few roots of largest modulus are not governed by this bulk behaviour, and the question asks how likely they are to be real. Part 3 is a persistence-type question: it asks how fast the probability decays that many consecutive extreme roots all lie on the real line.

What is known

• The expected number of real roots of $P_n$ is asymptotic to $\frac{2}{\pi}\log n$ (Kac, 1943).
• For independent identically distributed coefficients, the arguments of the roots are asymptotically uniformly distributed, and under a mild moment condition, which Gaussian coefficients satisfy, the roots concentrate near the unit circle (Ibragimov and Zaporozhets).
• The roots of largest and of smallest modulus behave non-universally: they do not converge to the unit circle, and the largest modulus is heavy tailed, with the number of its finite moments limited by the behaviour of the coefficient distribution near 0. The roots of modulus less than one converge to a limiting point process (Butez, 2018).
• The zeros of the random power series $f(z) = \sum_{j \ge 0} \xi_j z^j$ with independent real Gaussian coefficients form a Pfaffian point process (Matsumoto and Shirai, 2013), so their correlation functions are explicit.

What would count as progress

• A written proof that the limits $p(k)$ exist. Reversing the order of the coefficients does not change the law of $P_n$, so $p_n(k)$ is also the probability that the $k$ roots of smallest modulus are all real; since $P_n \to f$ uniformly on compact subsets of the unit disk, this part may follow from Hurwitz's theorem, with $p(k)$ the probability that the $k$ zeros of $f$ nearest the origin are real.
• A numerical estimate of $p(1)$ with rigorous error bars, for example from Monte Carlo simulation of $P_n$ for large $n$ or from the Pfaffian formulas for the zeros of $f$.
• Upper and lower bounds on the decay of $p(k)$ as $k \to \infty$, and then its exact asymptotic form.

References

1. M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314–320.
2. I. Ibragimov and D. Zaporozhets, On distribution of zeros of random polynomials in complex plane, Springer Proc. Math. Stat. (2012), 303–323; arXiv:1102.3517.
3. R. Butez, The largest root of random Kac polynomials is heavy tailed, Electron. Commun. Probab. 23 (2018), 1–9.
4. S. Matsumoto and T. Shirai, Correlation functions for zeros of a Gaussian power series and Pfaffians, Electron. J. Probab. 18 (2013), 1–18.

Source

Imported from AIM Problem Lists randpolyzero 1.4, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.