Consider a sequence of polynomials P_n(x), n=1,2,..., with only nonpositive zeros and with deg P_n <= n. The aim of this talk is to present a user-friendly approach for determining the limiting zero distribution of P_n as n -> \infty. The method is based on establishing an equivalence between the limiting empirical zero distribution of P_n and an exponential profile associated with the coefficients of the polynomials. Using this equivalence we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. The talk is based on a joint work with Jonas Jalowy and Alexander Marynych, ArXiV: 2504.11593.

This talk is based on a joint work with Stephen Muirhead. We establish the existence and uniqueness of a well-concentrated giant component in the supercritical excursion sets of three important ensembles of spherical Gaussian random fields: Kostlan's ensemble, band-limited ensembles, and the random spherical harmonics. Our main results prescribe quantitative bounds for the volume fluctuations of the giant that are essentially optimal for non-monochromatic ensembles, and suboptimal but still strong for monochromatic ensembles. Our results support the emerging picture that giant components in Gaussian random field excursion sets have similar large-scale statistical properties to giant components in supercritical Bernoulli percolation. The proofs employ novel decoupling inequalities for spherical ensembles which are of independent interest.

In this talk I will discuss the behavior of the expected number of real zeros, from leading order to full asymptotic expansion, of real random polynomials spanned by orthogonal polynomials on the unit circle with coefficients that are i.i.d. real normal random variables.

We investigate the random variable defined by the volume of the zero set of a smooth Gaussian field, on a general Riemannian manifold possibly with boundary. We prove a new explicit formula for its Wiener-Itô chaos decomposition that is notably simpler than existing alternatives and which holds in greater generality, without requiring the field to be compatible with the geometry of the manifold. A key advantage of our formulation is a significant reduction of the complexity of the computations of the variance of the nodal volume. Unlike the standard Hermite expansion, which requires evaluating the expectation of products of 2+2n Hermite polynomials, our approach reduces this task—in any dimension n—to computing the expectation of a product of just four Hermite polynomials. As a consequence, we establish a new exact formula for the variance. This is a joint work with Michele Stecconi.

This talk presents joint work with Yen Do and Sean O'Rourke on the real roots of random polynomials with non-centered coefficients, a natural but less understood setting. While Ibragimov and Maslova analyzed the expected number of real roots of Kac polynomials in the 1970s, key questions concerning the variance and central limit theorem have remained unresolved for over fifty years. We answer these questions and develop a unified method that applies to a broad class of non-centered random polynomials, including Kac and hyperbolic polynomials, their derivatives, and related generalizations.

It is well known that a random polynomial of degree n with i.i.d. coefficients has most of its roots clustering near the unit circle (at scale 1/n) as n grows large. Moreover, the roots tend to repel from one another. I will present a recent result showing that the minimal separation distance between the roots, when normalized by n^{-5/4}, converges in distribution to a non-trivial limit law.  Based on a joint work with Marcus Michelen.

Let f_n be a random polynomial of degree n with i.i.d. mean zero subgaussian coefficients. We prove a law of large numbers result for log|disc(f_n)| where disc(f_n) is the discriminant of the polynomial f_n. This quantifies the repulsion between the roots of f_n. This is based on joint work with Oren Yakir.

A result of Hoskins and Steinerberger established that very high derivatives of random polynomials with real mean 0, variance 1 independent and identically distributed roots are approximately Hermite polynomials, up to a Gaussian shift of the argument. In this talk we will consider two probabilistic extensions of this result: (1) A central limit theorem describing the fluctuations around this deterministic Hermite limit. (2) A generalized limit theorem dropping the finite variance condition to consider general collections of independent roots which don't satisfy the classic CLT. The proofs apply finite free probability theory (which has been successful in describing deterministic polynomials) to study these questions around fluctuations. Based on joint work with Octavio Arizmendi and Katsunori Fujie.

Power engineers and researchers model the electric grid with an undirected graph whose structure induces a system of polynomial equations known as the power flow equations. These power flow equations describe the physical relationship between nodal power and voltage in a graph and thus model the flow of electricity throughout a power grid. However, only the real solutions to this polynomial formulation are physically meaningful. Finding all such solutions are critical for methods that output grid stability metrics. Bounds on the number of real solutions based on the number of vertices of the graph are useful inputs as stopping criteria for algorithms that provably find all real solutions, such as homotopy continuation. However, only slight improvements over the Bezout bound have been discovered; empirical studies on typical grids show that the number of real solutions scales significantly slower. In an effort to align closer to typical grid behavior, standard Gaussian random coefficients are introduced to this system of polynomial equations to analyze generic conditions for grid parameters. In this talk, I will show how to use the Kac-Rice formula to determine the expected number of real solutions to the power flow equations. This results in a formula that scales slower than current best bounds and closer aligns to the number of solutions to the power flow equations seen in practice. I will finally pose questions on the application of new results in the field of random polynomial theory such as local universality and the central limit theorem.

We will discuss he Edgeworth expansion for random sums with general random coefficients. We show that, for most choices of the steps, the expansion holds up to any polynomial order. Most interestingly, we establish that the Edgeworth expansion applies to random walks with steps derived from Weyl polynomials. Precise estimates for the expectation and variance of the number of real roots of Weyl polynomials will be computed.

This talk will discuss Hilbert's sixteenth problem from a probabilistic perspective. The broad question is "What can be said about the limit cycles of a typical planar vector field?" We will give particular attention to the simple case of limit cycles of a perturbed linear center. When the (bivariate) perturbative polynomials in the ODE system have i.i.d. coefficients, the problem of studying limit cycle bifurcations reduces to the study of zeros of a certain random univariate polynomials with independent but not identically distributed coefficients. Namely, the coefficient variance is inversely proportional to degree. This falls at the critical case of the "generalized Kac polynomials", and determining the average (for distributions beyond the Gaussian setting) required a novel adaptation of universality methods. We will also discuss future research concerning perturbed Hamiltonian systems leading to the study of zeros of Melnikov functions. This is joint work with Manjunath Krishnapur and Oanh Nguyen.

Random meromorphic function (RMF), introduced by Kabluchko & Wigman, is a quotient of two independent Gaussian entire functions. Thanks to their translation–rotation invariance and ergodicity, RMF is a compelling model for planar random geometry. In an ongoing joint work with Itai Bar-Deroma & Mikhail Sodin, we investigate the closely related problems of component counts and percolation behaviour in RMF preimages of smooth Jordan curves. This talk will outline our current results, open questions, and conjectures.

Over many decades, mathematicians have learned much about the distribution of the roots of single variable polynomials whose coefficients are independent, identically distributed Normal random variables. In this talk I will discuss extending these ideas to random sums that aren't necessarily polynomials. They could be random Fourier series or sums using other basis functions. I will also discuss how things change when the random coefficients aren't iid Normal but are some other interesting distribution.

The number complex zeros of a system of sparse polynomial equations is an invariant from algebraic geometry that is linked to theory of mixed volumes in convex geometry. We present average number of real zeros of sparse polynomial systems with a similar geometric interpretation and obtain estimates on this quantity. The real case has a combinatorial nature unlike its convex geometric counterpart in complex case, and this brings a collection of open problems: this talk is meant to be an invitation to the participants of the workshop to this widely open field. Joint work with Josue Tonelli-Cueto, Mate Telek, and Peter Burgisser.

Polynomial chaos refers to a polynomial function evaluated on a sequence of i.i.d. random variables. These objects arise naturally in a wide range of probabilistic models — including random graphs, spin glass systems, and random matrix theory, just to name a few. They constitute a powerful and flexible framework, particularly for the polynomial approximation of general random fields. In this talk, we characterize all possible limits of sequences of polynomial chaoses with uniformly bounded degrees, thereby resolving a longstanding open question in the field. After presenting several natural settings in which such structures appear, we will outline key applications of our results and provide an overview of the main ideas behind the proofs\

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. We discuss one of the important geometric observables: critical points. We compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius. We observe a similar behaviour for the point process of critical points of isotropic stationary Gaussian fields.

We show for a large class of entire functions, f, that after proper rescaling, on compact sets, the derivatives of f converge to cosine, in particular their roots become evenly spaced. This proves a conjecture of Farmer and Rhoades [Trans. Amer. Math. Soc., 357(9):3789–3811, 2005] and Farmer [Adv. Math., 411:Paper No. 108781, 14, 2022] for our class of entire functions. A main ingredient of our proof is to show that high derivatives of high degree polynomials behave like Hermite polynomials, which we prove using the techniques from the newly developed field of finite free probability.

A fundamental problem harder for real polynomial systems than complex polynomial systems is root counting: All classical algorithms have worst-case exponential-time complexity. Restricting to sparse polynomial reduces the number of real roots and, for very sparse systems, enables significant speed-ups (although the problem is NP-hard for sufficiently many monomial terms). We consider the case of n polynomials in n variables where the total number of distinct exponent vectors is n+2, the coefficients are integers of absolute value at most H, and the exponent vectors do not lie in a common affine hyperplane: The so-called circuit case. We outline an approach to attaining polynomial-time real root counting for circuit systems, for a fraction of coefficient values tending to 1 as H tends to infinity. The key idea is to prove a new, randomized version of a classical Diophantine approximation result of Baker and Wustholz. This builds upon recent joint work with Weixun Deng, Alperen Ergur, and Grigoris Paouris.

We show that the probability that a random polynomial with iid coefficients and even degree has no real roots decays polynomially in the degree with a universal persistence exponent, as soon as the iid coefficients have mean zero and variance one. Our results apply to a broader class of polynomials, where the coefficients can be multiplied by a regularly varying function. This talk is based on joint work with Promit Ghosal at University of Chicago.

I will discuss the evolution of the complex roots of random polynomials under differential flows, focusing on two main examples: repeated differentiation and the heat flow. In both cases, I will describe how the roots evolve in time and highlight connections with random matrix theory.

Free probability is a non-commutative version of (classic) probability that has applications to random matrices. Interestingly enough, free probability can be used to study the asymptotic root distribution of certain basic operations with (deterministic) polynomials. The bridge between these two topics is known as finite free probability, a topic started less than a decade ago when Marcus, Spielman and Srivastava noticed that the expected characteristic polynomial of randomly rotated matrices can be understood using some classical operations on polynomials, studied a century ago by Szegö and Walsh. The main goal of this talk is to survey the areas of free probability and finite free probability, including the connection between: -repeated differentiation and free fractional convolution. -ratio of consecutive coefficients and Voiculescu's S-transform. Time permitting we will discuss an ongoing project on the study of repeated polar differentiation.

If p(z) is a non-constant complex polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when p(z) is a random polynomial of degree n with roots chosen independently from a radially symmetric, compactly supported probably measure mu in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of p(z). This allows us to compute the asymptotic fluctuations of the largest critical points as the degree n tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of mu near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials.