On the Expected Number of Real Roots of a System of Random Polynomial Equations

On Classifications of Random Polynomials

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...

On the average number of sharp crossings of certain Gaussian random polynomials

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

The methods of Shub and Smale [SS93] are extended to the class of multihomogeneous systems of polynomial equations, yielding Theorem 1, which is a formula expressing the mean (with respect to a particular distribution on the space of coefficient vectors) number of real roots as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form expressions...

Expected Number of Local Maxima of Some Gaussian Random Polnomials

2 00 6 Average volume , curvatures , and Euler characteristic of random real algebraic varieties

We determine the expected curvature polynomial of random real projec-tive 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 t...