Random polynomials with prescribed Newton polytope

By Bernard Shiffman, Steve Zelditch. 2002

60 citations recorded by the source corpus.

The Newton polytope Pf of a polynomial f is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of m polynomials with given Newton polytopes. In this article, we show that Pf also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degreeN in m complex variables with its usual SU(m + 1)-invariant Gaussian measure and then consider the conditional measures |NP induced on the subspace of polynomials whose Newton polytope PfNP. When P = �, the unit simplex, then it is obvious and well-known that the expected distribution of zeros Zf1,...,fk (regarded as a current) of k polynomials f1,...,fk of degree N is uniform relative to the Fubini-Study form. Our main results concern the conditional expectation E|NP(Zf1,...,fk ) of zeros of k polynomials with Newton polytope NPNp� (where p = deg P); these results are asymptotic as the scaling factor N ! 1. We show that E|NP(Zf1,...,fk ) is asymptotically uniform on the inverse image AP of the open scaled polytope 1 P ◦ via the moment map µ : CP m ! � for projective space. However, the zeros have an exotic distribution outside of AP and when k = m (the case of the Kouchnirenko-Bernstein theorem) the percentage of zeros outside AP approaches 0 as N ! 1.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4eeb881af7819d2044fbee49e30574abef81de5a

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Citations in the corpus

ZEROS OF RANDOM POLYNOMIALS ON C m (2006, background): “ zero current of kindependent random functions f 1,...,f k ∈ F is given by E(Z f1,...,f k ) = i 2π ∂∂¯logS F(z,z) k . The proof follows by a verbatim repetition of the proof of Proposition 5.1 in [SZ3] (which gives the case where the g j are normalized monomials with exponents in a Newton polytope”
Dimer models from mirror symmetry and quivering amoebae (2005, background): “hip between the number of solutions of polynomial systems and the Newton polytope spanned by the polynomials, culminating in the Koushnirenko-Bernstein theorem [48, 49] (for relations to amoebæ, q.v. [21, 50]): The number of joint zeros in (C∗)m of m generic polynomials {f 1,f2,...f m} with a given ”
Equidistribution of zeros of random holomorphic sections (2013, unknown): “If a (n) j are standard complex Gaussian then it is classical that (see [26, 27, 7]) the identity (3.”