Random polynomials with prescribed Newton polytope, I

Details Random polynomials with prescribed Newton polytope, I Random polynomials with prescribed Newton polytope, I

2002-03-08

arxiv.org/abs/math/0203074v2

Mathematics

Algebraic Geometry

Changes to introduction and other minor changes; 56 pages, 7 figures

Scientific paper

The Newton polytope $P_f$ 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 $P_f$ also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degree $\leq N$ in $m$ complex variables with its usual $SU(m+1)$-invariant Gaussian measure and then consider the conditional measures $\gamma_{|NP}$ induced on the subspace of polynomials whose Newton polytope $P_f\subset NP$. When $P=\Sigma$, the unit simplex, then it is obvious and well-known that the expected distribution of zeros $Z_{f_1,...,f_k}$ (regarded as a current) of $k$ polynomials $f_1,...,f_k$ of degree $N$ is uniform relative to the Fubini-Study form. Our main results concern the conditional expectation $E_{|NP} (Z_{f_1,...,f_k})$ of zeros of $k$ polynomials with Newton polytope $NP\subset Np\Sigma$ (where $p=\deg P$); these results are asymptotic as the scaling factor $N\to\infty$. We show that $E_{|NP} (Z_{f_1,...,f_k})$ is asymptotically uniform on the inverse image $A_P$ of the open scaled polytope $p^{-1}P^\circ$ via the moment map $\mu:CP^m\to\Sigma$ for projective space. However, the zeros have an exotic distribution outside of $A_P$ and when $k=m$ (the case of the Kouchnirenko-Bernstein theorem) the percentage of zeros outside $A_P$ approaches 0 as $N\to\infty$.

Affiliated with

Also associated with

No associations

Rating

Random polynomials with prescribed Newton polytope, I does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Random polynomials with prescribed Newton polytope, I, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random polynomials with prescribed Newton polytope, I will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-268429