Random polynomials with prescribed Newton polytope, I

الأصل · EN

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 degree ≤ N in m complex variables with its usual SU(m+1)-invariant Gaussian measure and then consider the conditional measures γ|NP induced on the subspace of polynomials whose Newton polytope Pf⊂ NP. When P=Σ, 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 NP⊂ NpΣ (where p=°P); these results are asymptotic as the scaling factor N→∞. We show that E|NP (Zf₁,...,fₖ) is asymptotically uniform on the inverse image Aₚ of the open scaled polytope p⁻¹P∘ via the moment map μ:CPᵐ→Σ 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→∞.