Lars Simon
We prove several results on homogeneous plurisubharmonic polynomials on $\mathbb{C}^n$, $n\in\mathbb{Z}_{\geq 2}$. Said results are relevant to the problem of constructing local bumpings at boundary points of pseudoconvex domains of finite D'Angelo $1$-type in $\mathbb{C}^{n+1}$.
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Each extreme edge of the Newton diagram of a plurisubharmonic polynomial on $\mathbb{C}^2$ gives rise to a plurisubharmonic polynomial. It is tempting to believe that the union of the extreme edges or the convex hull of said union will do the same. We construct a plurisubharmonic polynomial $P$ on $\mathbb{C}^2$ with precisely two extreme edges $E_1$ and $E_2$, such that neither $E_1\cup{E_2}$ nor $\text{Conv}({E_1\cup{}E_2})$ yields a plurisubharmonic polynomial.
We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain Ω\subset C^n in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with bdy(Ω), at the site of the bumping, are explicitly realised. Generally, when Ω\subset C^n, n\geq 3, the known methods lead to bumpings with high orders of contact -- which are not explicitly known either -- at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in C^3. …
The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of polynomial, algebraic and analytic functions. The paper has much in common with an earlier paper of the author.
Let T be a positive plurisubharmonic current of bidimension (p,p) and let $δ>0$. Assume that the Lelong number of T satisfies $ν(T,a)\geq δ$ on a dense subset of supp(T) (rectifiable currents satisfy this condition). Then $T=ϕ[X]$, where X is a complex subvariety of pure dimension p and $ϕ$ is a weakly plurisubharmonic function on X. We have an analogous result for plurisuperharmonic currents. We also introduce and study a notion of polynomial p-hull.
We say that a symmetric noncommutative polynomial in the noncommutative free variables (x_1, x_2, ..., x_g) is noncommutative plurisubharmonic on a noncommutative open set if it has a noncommutative complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size. In this paper, we show that if a noncommutative polynomial is noncommutative plurisubharmonic on a noncommutative open set, then the polynomial is actually noncommutative plurisubharmonic everywhere and has the form p = \sum f_j^T f_j + \sum k_j k_j^T + F + F^T where the sums are f…
We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the symmetric noncommutative plurisubharmonic polynomials are precisely the noncommutative convex polynomials with a noncommutative analytic change of variables; i.e., a noncommutative symmetric polynomial…
We correct the calculation of the Monge-Ampère measure of a certain extremal plurisubharmonic function for the complex Euclidean ball in C^2.
Log canonical thresholds at infinity
本文研究了 plurisubharmonic 函数在无穷远处的 log canonical 阈值问题，通过 toric 情况下的显式公式和多项式逼近方法，探讨了 $e^{-u} \in L^p$ 的范围及 tame 多项式映射的应用。
Let $K$ be a compact subset of a totally-real manifold $M$, where $M$ is either a $\mathcal{C}^2$-smooth graph in $\mathbb{C}^{2n}$ over $\mathbb{C}^n$, or $M=u^{-1}\{0\}$ for a $\mathcal{C}^2$-smooth submersion $u$ from $\mathbb{C}^n$ to $\mathbb{R}^{2n-k}$, $k\leq n$. In this case we show that $K$ is polynomially convex if and only if for a fixed neighbourhood $U$, defined in terms of the defining functions of $M$, there exists a plurisubharmonic function $Ψ$ on $\mathbb{C}^n$ such that $K\subset \{Ψ<0\}\subset U$.
Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results
本文综述指数限制在紧凸集mS内的分次多项式环P^S(C^n)相关多下调和理论，研究其与增长性由S对数支撑函数决定的全局多下调和函数类L^S(C^n)的关系，介绍二者加权极值函数的性质及与S性质的关联。
Let $f$ be a Hénon-Sibony map (regular polynomial automorphism) of $\mathbb{C}^k$ and let $μ$ be the equilibrium measure of $f$. In this paper we prove that $μ$ is exponentially mixing for plurisubharmonic test functions.
We prove the existence of canonical tubular neighbourhoods around complex submanifolds of Kähler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Ampère equation on the deformation to the normal cone of the submanifold. We use this to establish local regularity for global weak solutions, giving local smoothness to the (weak) geodesic rays in the space of (weak) Kähler potentials associated to a given complex submanifold. We also use it to get an optimal regularity result for naturally defined plurisubharmonic env…
The Sidon constant for the index set of nonzero m-homogeneous polynomials P in n complex variables is the supremum of the ratio between the l^1 norm of the coefficients of P and the supremum norm of P in D^n. We present an estimate which gives the right order of magnitude for this constant, modulo a factor depending exponentially on m. We use this result to show that the Bohr radius for the polydisc D^n is bounded from below by a constant times sqrt((log n)/n).
We give a criterion which characterizes a homogeneous real multi-variate polynomial to have the property that all sufficiently large powers of the polynomial (as well as their products with any given positive homogeneous polynomial) have positive coefficients. Our result generalizes a result of De Angelis, which corresponds to the case of homogeneous bi-variate polynomials, as well as a classical result of Pólya, which corresponds to the case of a specific linear polynomial. As an application, we also give a characterization of certain polynomial beta functions, which are the spectral radius f…
Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.
Minor modifications are given to prove the Main Theorem under the Blaschke (instead of Carleson) condition as well as a small historical comment.
The main goal of this paper is to prove the Schmidt--Kolchin conjecture. This conjecture says the following: the vector space of degree \(d\) differentially homogeneous polynomials in \((N+1)\) variables is of dimension \((N+1)^{d}\). Next, we establish a one-to-one correspondance between differentially homogeneous polynomials in \((N+1)\) variables, and twisted jet differentials on projective spaces. As a by-product of our study of differentially homogeneous polynomials, we are thus able to understand explicitly twisted jet differentials on projective spaces.
On polyharmonic polynomials
本文研究齐次多项式到齐次多重调和多项式空间的正交投影，通过基本解导数的 Kelvin 变换给出多项式分解，并讨论向量基与正交级数收敛性。
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