Average volume, curvatures, and Euler characteristic of random real algebraic varieties

We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective varieties is found. This considerably extends previously known results on the number of roots, the volume, and the Euler characteristic of the solution set of random polynomial equationsComment: 38 pages. Version 2: corrected typos, changed some notation, rewrote proof of Theorem 5.

On the number of minima of a random polynomial

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain the bound O(exp(-beta n^2 + (n/2) log (d-1))) (beta is a positive constant independent on n and d) for the number of minima of such a polynomial. This proves that most normal random polynomials of fixed degree have only saddle points. Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions.Comment: 22 pages. We learned since the first version that the probability that a matrix in GOE(n) is positive definite is known. This follows from the theory of large deviations (reference in the paper). Therefore, we can now state a precise upper bound (Theorem 2) for the number of minima of a random polynomial, instead of a bound depending on that probabilit

The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations

Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form expressions for the mean in special cases that include earlier results of Shub and Smale (for the general homogeneous system) and Rojas (for ``unmixed'' multihomogeneous systems). Theorem 3 gives upper and lower bounds for the mean number of roots, where the lower bound is the square root of the generic number of complex roots, as determined by Bernstein's theorem. These bounds are derived by induction from recursive inequalities given in Theorem 4

Limit cycle enumeration in random vector fields

We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from the Kostlan-Shub-Smale ensemble. For the related Bargmann-Fock ensemble of real analytic functions we establish an asymptotic result for the average number of empty limit cycles (limit cycles that do not surround other limit cycles) in a large viewing window. Concerning the special setting of limit cycles near a randomly perturbed center focus (where the perturbation has i.i.d. coefficients) we prove that the number of limit cycles situated within a disk of radius less than unity converges almost surely to the number of real zeros of a certain random power series. We also consider infinitesimal perturbations where we obtain precise asymptotics on the global count of limit cycles for a family of models. The proofs of these results use novel combinations of techniques from dynamical systems and random analytic functions.Comment: 37 pages. This version includes more details in the proofs, an expanded section on preliminaries, a new section on infinitesimal perturbations, and a concluding remarks section with open problems and future directions. The statement of the conjecture in the introduction has been slightly modifie