Random polynomials with prescribed Newton polytope

The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its behavior. The Bernstein-Kouchnirenko Theorem asserts that even the number of simultaneous zeros in $(\mathbb{C}^*)^m$ of a system of $m$ polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes also have a strong impact on the distribution of zeros and pointwise norms of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in $(\mathbb{C}^*)^m$ for these distributions.

Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of 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 $\mathbf{E}_{\vert N P}(Z_{f_1, \dots, f_k})$ of simultaneous zeros of $k$ polynomials with Newton polytope $NP$ as $N \to \infty$. When $P = \Sigma$, the unit simplex, it is clear that the expected zero distributions $\mathbf{E}_{\vert N\Sigma}(Z_{f_1, \dots, f_k})$ are uniform relative to the Fubini-Study form. For a convex polytope $P\subset p\Sigma$, we show that there is an allowed region on which $N^{-k}\mathbf{E}_{\vert N P}(Z_{f_1, \dots, f_k})$ 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 case of the Bernstein-Kouchnirenko Theorem), the expected percentage of simultaneous zeros in the forbidden region approaches 0 as $N\to\infty$.