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Bernard Shiffman

Publications and source records attributed to Bernard Shiffman.

33 records · Page 2Linked to original sources
arXiv2003

Random polynomials with prescribed Newton polytope

We show that the Newton polytope of a polynomial has a strong impact on the distribution of its mass and zeros. The basic theme is that Newton polytopes determine allowed and forbidden regions for these distributions. We equip the space of (holomorphic) polynomials of degree $\leq p$ in $m$ complex variables with its usual $SU(m + 1)$-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope $P$. We then determine the asymptotics of the conditional expectation $E_{|N P}(Z^k)$ of simultaneous zeros of $k$ polynomials with Newton polytope $NP$ as $N \to \infty$. When $P = Σ$, the unit simplex, then it is obvious and well-known that the expected zero distribution $E_{|NΣ}(Z^k)$ is uniform relative to the Fubini-Study form. For a convex polytope $P\subset pΣ$, we show that there is an allowed region on which $N^{-k}E_{|N P}(Z^k)$ is asymptotically uniform as the scaling factor $N\to\infty$. However, the zeros have an exotic distribution in the complementary forbidden region, and when $k=m$, the expected percentage of simultaneous zeros in the forbidden region approaches 0 as $N\to\infty$, yielding a quantitative version of the Kouchnirenko-Bernstein theorem.

math.AG↗
arXiv2002

Equilibrium distribution of zeros of random polynomials

We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain $Ω\subset C$ relative to an analytic weight $ρ(z) |dz|$. In the simplest case where $Ω$ is the unit disk and $ρ=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is the uniform measure on $S^1$. We show that for any analytic $(Ω, ρ)$, the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on $\partialΩ$ as $N\to\infty$. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of $(Ω, ρ)$.

math.CV↗
arXiv2002

Random polynomials with prescribed Newton polytope, I

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 $N\to\infty$.

math.AG↗
arXiv2002

Constructing low degree hyperbolic surfaces in P^3

We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a "hyperbolic non-percolation" property. We use this method to show that general small deformations of certain singular abelian surfaces of degree 8 are hyperbolic. We also show that a union of 15 planes in general position in projective 3-space admits hyperbolic deformations.

math.AG↗
arXiv2001

Hyperbolic hypersurfaces in P^n of Fermat-Waring type

In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in P^n of degree 4(n-1)^2, for all dimensions n>1. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P^n of degree 4n^2-2n+1 possessing complete hyperbolic, hyperbolically embedded complements.

math.AG↗
arXiv2000

Correlations between zeros and supersymmetry

In our previous work [math-ph/9904020], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions.

math-ph↗
arXiv2000

Universality and scaling of zeros on symplectic manifolds

This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\to\infty$. We fix a point $P$ and focus on the ball of radius $1/\sqrt{N}$ about $P$. Under a microscope magnifying the ball by the factor $\sqrt{N}$, the statistics of the configurations of simultaneous zeros of random $k$-tuples of sections tends to a universal limit independent of $P,M,L$. We review this result and generalize it further to the case of pre-quantum line bundles over almost-complex symplectic manifolds $(M,J,ω)$. Following [SZ2], we replace $H^0(M,L^N)$ in the complex case with the `asymptotically holomorphic' sections defined by Boutet de Monvel-Guillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an $m$-dimensional setting of the Kac-Rice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random $k$-tuples of asymptotically holomorphic sections belong to the same universality class as in the complex case.

math-ph↗
arXiv2000

Random almost holomorphic sections of ample line bundles on symplectic manifolds

The spaces $H^0(M, L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact Kähler manifold $(M,ω)$ have been generalized by Boutet de Monvel and Guillemin to spaces $H^0_J(M, L^N)$ of `almost holomorphic sections' of ample line bundles over an almost complex symplectic manifold $(M, J, ω)$. We consider the unit spheres $SH^0_J(M, L^N)$ in the spaces $H^0_J(M, L^N)$, which we equip with natural inner products. Our purpose is to show that, in a probabilistic sense, almost holomorphic sections behave like holomorphic sections as $N \to \infty$. Our first main result is that almost all sequences of sections $s_N \in SH^0_J(M, L^N)$ are `asymptotically holomorphic' in the Donaldson-Auroux sense that $||s_N||_{\infty}/||s_N||_{2} = O(\sqrt{\log N})$, $||\bar{\partial} s_N||_{\infty}/||s_N||_{2} = O(\sqrt{\log N})$ and $||\partial s_N||_{\infty}/||s_N||_{2} = O(\sqrt{N \log N})$. Our second main result concerns the joint probability distribution of the random variables $s_N(z^p),\ \nabla s_N(z^p)$, $1\le p\le n$, for $n$ distinct points $z^1,..., z^n$ in a neighborhood of a point $P_0\in M$. We show that this joint distribution has a universal scaling limit about $P_0$ as $N \to \infty$. In particular, the limit is precisely the same as in the complex holomorphic case. Our methods involve near-diagonal scaling asymptotics of the Szegö projector $Π_N$ onto $H^0_J(M, L^N)$, which also yields proofs of symplectic analogues of the Kodaira embedding theorem and Tian asymptotic isometry theorem.

math.SG↗
arXiv1999

Universality and scaling of correlations between zeros on complex manifolds

We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ of a positive holomorphic line bundle $L$ over a compact complex manifold $M$, when distances are rescaled so that the average density of zeros is independent of $N$. We show that the limit correlation is independent of the line bundle and depends only on the dimension of $M$ and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay's limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.

math-ph↗
arXiv1999

Poincare-Lelong approach to universality and scaling of correlations between zeros

This note is concerned with the scaling limit as N approaches infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L^N of any positive holomorphic line bundle L over a compact Kahler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the "correlation forms" are universal, i.e. independent of the bundle L, manifold M or point on M.

math-ph↗
arXiv1998

Two classes of hyperbolic surfaces in P^3

We construct two classes of singular Kobayashi hyperbolic surfaces in $P^3$. The first consists of generic projections of the cartesian square $V = C \times C$ of a generic genus $g \ge 2$ curve $C$ smoothly embedded in $P^5$. These surfaces have C-hyperbolic normalizations; we give some lower bounds for their degrees and provide an example of degree 32. The second class of examples of hyperbolic surfaces in $P^3$ is provided by generic projections of the symmetric square $V' = C_2$ of a generic genus $g \ge 3$ curve $C$. The minimal degree of these surfaces is 16, but this time the normalizations are not C-hyperbolic.

math.AG↗
arXiv1998

Distribution of zeros of random and quantum chaotic sections of positive line bundles

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers $L^N$ of a positive holomorphic Hermitian line bundle $L$ over a compact complex manifold $M$. Our first result concerns `random' sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$, we show that for almost every sequence $\{S^N_j\}$, the associated sequence of zero currents $1/N Z_{S^N_j}$ tends to the curvature form $ω$ of $L$. Thus, the zeros of a sequence of sections $s_N \in H^0(M, L^N)$ chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$ consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

math.CV↗
arXiv1996

Value distribution for sequences of rational mappings and complex dynamics

We obtain results on the asymptotic equidistribution of the pre-images of linear subspaces for sequences of rational mappings between projective spaces. As an application to complex dynamics, we consider the iterates $P_k$ of a rational mapping $P$ of $\PP^n$. We show, assuming a condition on the topological degree $λ$ of $P$, that there is a probability measure $μ$ on $\PP^n$ such that the discrete measures $λ^{-k}P_k^*δ_w$ converge to $μ$ for all $w\in\PP^n$ outside a pluripolar set.

math.CV↗
arXiv1993

Algebraic approximations of holomorphic maps from Stein domains to projective manifolds

It is shown that every holomorphic map $f$ from a Runge domain $Ω$ of an affine algebraic variety $S$ into a projective algebraic manifold $X$ is a uniform limit of Nash algebraic maps $f_ν$ defined over an exhausting sequence of relatively compact open sets $Ω_ν$ in $Ω$. A relative version is also given: If there is an algebraic subvariety $A$ (not necessarily reduced) in $S$ such that the restriction of $f$ to $A\capΩ$ is algebraic, then $f_ν$ can be taken to coincide with $f$ on $A\capΩ_ν$. The main application of these results, when $Ω$ is the unit disk, is to show that the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric of a quasi-projective algebraic manifold $Z$ are computable solely in terms of the closed algebraic curves in $Z$. Similarly, the $p$-dimensional Eisenman metric of a quasi-projective algebraic manifold can be computed in terms of the Eisenman volumes of its $p$-dimensional algebraic subvarieties. Another question addressed in the paper is whether the approximations $f_ν$ can be taken to have their images contained in affine Zariski open subsets of $X$. By using complex analytic methods (pluricomplex potential theory and Hörmander's $L^2$ estimates), we show that this is the case if $f$ is an embedding (with $\dim S<\dim X$) and if there is an ample line bundle $L$ on $X$ such that

alg-geom↗