On the average number of real roots of a random algebraic equation

Lyapunov Exponents for Particles Advected in Compressible Random Velocity Fields at Small and Large Kubo Numbers

“…An approximate expression for the Lyapunov exponent can be obtained by averaging the gradient A at the minima, that is at the stagnation points of the flow velocity u with A < 0. The distribution of A at u = 0 can be estimated using the Kac-Rice formula [31,32]:…”

“…In this case the particles are predominantly found near stagnation points of the flow (where the flow velocity vanishes) with negative velocity gradients, that is near the minima of the corresponding potential function. In this regime the Lyapunov exponent is determined by the flow-gradient fluctuations near these points, and we show how to compute the exponent using the KacRice formula [31,32] for counting singular points of random functions.…”

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

“…This line of investigation originates in the groundbreaking work of M. Kac [23] and S.O. Rice [37] who studied the distribution of zeros of certain random functions of one real variable.…”

Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials

“…In a seminal paper Littlewood and Offord [11] prove that the number of real roots of a p ∈ L n , on average, lies between c 1 log n log log log n and c 2 log 2 n and it is proved by Boyd [7] that every p ∈ L n has at most c log 2 n/ log log n zeros at 1 (in the sense of multiplicity). Kac [10] shows that the expected number of real roots of a polynomial of degree n with random uniformly distributed coefficients is asymptotically (2/π) log n. He writes "I have also stated that the same conclusion holds if the coefficients assume only the values 1 and −1 with equal probabilities. Upon closer examination it turns out that the proof I had in mind is inapplicable....…”