Random polynomials with prescribed Newton polytope, I
| dc.creator | Shiffman, Bernard | |
| dc.creator | Zelditch, Steve | |
| dc.date | 2002-03-08 | |
| dc.date | 2002-03-12 | |
| dc.date.accessioned | 2026-07-07T04:46:54Z | |
| dc.date.available | 2026-07-07T04:46:54Z | |
| dc.description | 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 $γ_{|NP}$ induced on the subspace of polynomials whose Newton polytope $P_f\subset NP$. When $P=Σ$, 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 $NP\subset NpΣ$ (where $p=°P$); 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 $μ:CP^m\toΣ$ 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 $N\to\infty$. | |
| dc.description | Changes to introduction and other minor changes; 56 pages, 7 figures | |
| dc.identifier | https://arxiv.org/abs/math/0203074 | |
| dc.identifier | http://arxiv.org/abs/math/0203074 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63519 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Complex Variables | |
| dc.subject | Probability | |
| dc.title | Random polynomials with prescribed Newton polytope, I | |
| dc.type | text |