Random polynomials with prescribed Newton polytope

Shiffman, Bernard; Zelditch, Steve

Random polynomials with prescribed Newton polytope

Authors

Description

We show that the Newton polytope of a polynomial has a strong impact on the distribution of its mass and zeros. The basic theme is that Newton polytopes determine allowed and forbidden regions for these distributions. We equip the space of (holomorphic) polynomials of degree $\leq p$ 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 $E_{|N P}(Z^k)$ of simultaneous zeros of $k$ polynomials with Newton polytope $NP$ as $N \to \infty$. When $P = Σ$, the unit simplex, then it is obvious and well-known that the expected zero distribution $E_{|NΣ}(Z^k)$ is uniform relative to the Fubini-Study form. For a convex polytope $P\subset pΣ$, we show that there is an allowed region on which $N^{-k}E_{|N P}(Z^k)$ is asymptotically uniform as the scaling factor $N\to\infty$. However, the zeros have an exotic distribution in the complementary forbidden region, and when $k=m$, the expected percentage of simultaneous zeros in the forbidden region approaches 0 as $N\to\infty$, yielding a quantitative version of the Kouchnirenko-Bernstein theorem.
Revised exposition: terminology and facts concerning toric varieties have been moved to an appendix. This posting supersedes our article of similar title posted on the archive as math.AG/0203074. By using Euler-Maclaurin sum formulae for polytopes instead of Toeplitz analysis on toric varieties, we significantly simplied the proofs and generalized the results