Random polynomial systems and balanced metrics on toric varieties

Thursday, April 26, 2007 - 10:15am - 11:10am
Keller 3-180
J. Maurice Rojas (Texas A & M University)
Suppose c0,...,cd are independent identically distributed real Gaussians with mean 0 and variance 1. Around the 1940s, Kac and Rice proved that the expected number of real roots of the polynomial c0 + c1 x + ... + cd xdi, then the expected number of real roots is EXACTLY the square root of d. Aside from the cute square root phenomenon, Kostlan also observed that the distribution function of the real roots is constant with respect to the usual metric on the real projective line.

The question of what a natural probability measure for general multivariate polynomials then arises. We exhibit two (equivalent) combinatorial constructions that conjecturally yield such a measure. We show how our conjecture is true in certain interesting special cases, thus recovering earlier work of Shub, Smale, and McLennan. We also relate our conjecture to earlier asymptotic results of Shiffman and Zelditch on random sections of holomorphic line bundles.

This talk will deal concretely with polynomials and Newton polytopes, so no background on probability or algebraic geometry is assumed.