Random polynomials with prescribed Newton polytope
Bernard Shiffman, Steve Zelditch · Journal of the American Mathematical Society · 2003
The Newton polytope P f P_f of a polynomial f 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 ( C ∗ ) m (\mathbb {C}^*)^m of a system of m 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 ( C ∗ ) m (\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 ≤ p \leq p in m m complex variables with its usual SU ( m + 1 ) (m+1) -invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope P P . We then determine the asymptotics of the conditional expectation E | N P ( Z f 1 , … , f k ) \mathbf {E}_{|N P}(Z_{f_1, \dots , f_k}) of simultaneous zeros of