Random Polynomials with Prescribed Newton Polytope

The Newton polytope Pf of a polynomial f is well known to have a strong impact on its behavior. The Kouchnirenko-Bernstein theorem asserts that even the number of simultaneous zeros in (C∗)m of a system of m polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes further have a strong impact on the distribution of mass and zeros of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in (C∗)m for these distributions. Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of (holomorphic) polynomials of degree ≤ p in m complex variables with its usual SU(m+1)-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope P . We then determine the asymptotics of the conditional expectation E|NP (Zf1,...,fk ) of simultaneous zeros of k polynomials with Newton polytope NP as N → ∞. When P = Σ, the unit simplex, then it is obvious and well-known that the expected zero distribution E|NΣ(Zf1,...,fk ) is uniform relative to the Fubini-Study form. For a convex polytope P ⊂ pΣ, we show that there is an allowed region on which NE|NP (Zf1,...,fk ) is asymptotically uniform as the scaling factor N → ∞. However, the zeros have an exotic distribution in the complementary forbidden region and when k = m (the case of the KouchnirenkoBernstein theorem), the expected percentage of simultaneous zeros in the forbidden region approaches 0 as N → ∞.