Polynomial Hulls Research Articles

Holomorphic motions and polynomial hulls

A holomorphic motion of E ⊂ C E \subset \mathbb {C} over the unit disc D D is a map f : D × C → C f:D \times \mathbb {C} \to \mathbb {C} such that f ( 0 , w ) = w , w ∈ E f(0,w) = w,w \in E , the function f ( z , w ) = f z ( w ) f(z,w) = {f_z}(w) is holomorphic in z z , and f z : E → C {f_z}:E \to \mathbb {C} is an injection for all z ∈ D z \in D . Answering a question posed by Sullivan and Thurston [13], we show that every such f f can be extended to a holomorphic motion F : D × C → C F:D \times \mathbb {C} \to \mathbb {C} . As a main step a "holomorphic axiom of choice" is obtained (concerning selections from the sets C ∖ f z ( E ) , z ∈ D ) \mathbb {C}\backslash {f_z}(E),z \in D) . The proof uses earlier results on the existence of analytic discs in the polynomial hulls of some subsets of C 2 {\mathbb {C}^2} .

Polynomial hull of a sphere imbedded in C2

Polynomial hull of a sphere imbedded in C2

Polynomial hulls with convex fibers and complex geodesics

Let X be a compact subset of {| z | = 1} × C n with convex fibers. Several equivalent conditions are obtained that characterize polynomially convex hulls Y = X ̂ with nonempty interior. Under the latter assumption, it is shown that every ( z 0 , w 0 ) ϵ ∂ Y and such that ¦z 0 ¦ < 1 is contained in a closed n -dimensional complex submanifold of d × C n which is tangent to Y along an analytic disc. We show, as an application, that some results of Lempert on complex geodesies in convex domains are direct consequences of properties of polynomial hulls.

Polynomial hulls of sets with intervals as fibers

Polynomial hulls of sets with intervals as fibers

The Polynomial Hull of Non-Connected Tube Domains, and an Example of E. Kallin

The Polynomial Hull of Non-Connected Tube Domains, and an Example of E. Kallin

Polynomial Hulls of Sets Invariant Under an Action of the Special Unitary Group

If K is a compact subset of Cn, will denote the polynomial hull of K: arises in the study of uniform algebras as the maximal ideal space of the algebra P(K) of uniform limits on K of polynomials (see [3]). The condition (K is polynomially convex) is a necessary one for uniform approximation on K of continuous functions by polynomials (P(K) = C(K)). If K is not polynomially convex, the question of existence of analytic structure in is of particular interest. For n = 1, is the union of K and the bounded components of C\K. The determination of in dimensions greater than one is a more difficult problem. Among the special classes of compact sets K whose polynomial hulls have been determined are those invariant under certain group actions on Cn.

The Polynomial Hull of a Rectifiable Curve in C n

The Polynomial Hull of a Rectifiable Curve in C n

Envelopes of holomorphy and polynomial hulls

Envelopes of holomorphy and polynomial hulls

Regularity of varieties in strictly pseudoconvex domains

We prove a theorem on the boundary regularity of a purely p-dimensional complex subvariety of a relatively compact, strictly pseudoconvex domain in a Stein manifold. Some applications describing the structure of the polynomial hull of closed curves in Cn are also given.

The polynomial hull of a set of finite linear measure in Cn

The polynomial hull of a set of finite linear measure in Cn

An Analytic Set-Valued Selection and Its Applications to the Corona Theorem, to Polynomial Hulls and Joint Spectra

An Analytic Set-Valued Selection and Its Applications to the Corona Theorem, to Polynomial Hulls and Joint Spectra

Polynomial Hulls with Convex Sections and Interpolating Spaces

Assume that $L \subset \partial D \times {{\mathbf {C}}^m}$ is compact and has convex vertical sections. Denote by $K$ its polynomially convex hull. It is shown that $K\backslash \partial D \times {{\mathbf {C}}^m}$, if nonempty, can be covered by graphs of analytic functions $f:D \to {{\mathbf {C}}^m}$. The proof is based on complex interpolation theory for families of finite-dimensional normed spaces.

Polynomial hulls with convex sections and interpolating spaces

Assume that L ⊂ ∂ D × C m L \subset \partial D \times {{\mathbf {C}}^m} is compact and has convex vertical sections. Denote by K K its polynomially convex hull. It is shown that K ∖ ∂ D × C m K\backslash \partial D \times {{\mathbf {C}}^m} , if nonempty, can be covered by graphs of analytic functions f : D → C m f:D \to {{\mathbf {C}}^m} . The proof is based on complex interpolation theory for families of finite-dimensional normed spaces.

An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra

It is shown that for every annulus P = { z ∈ C n : δ &gt; | z | &gt; r } P = \{ z \in {{\mathbf {C}}^n}:\delta &gt; |z| &gt; r\} , δ &gt; 0 \delta &gt; 0 , there exists a set-valued correspondence z → K ( z ) : P → 2 C n z \to K(z):P \to {2^{{{\mathbf {C}}^n}}} , whose graph is a bounded relatively closed subset of the manifold { ( z , w ) ∈ P × C n : z 1 w 1 + ⋯ + z n w n = 1 } \{ (z,w) \in P \times {{\mathbf {C}}^n}:{z_1}{w_1} + \cdots + {z_n}{w_n} = 1\} which can be covered by n n -dimensional analytic manifolds. The analytic set-valued selection K K obtained thereby is then applied to several problems in complex analysis and spectral theory which involve solving the equation a 1 x 1 + ⋯ + a n x n = y {a_1}{x_1} + \cdots + {a_n}{x_n} = y . For example, an elementary proof is given of the following special case of a theorem due to Oka: every bounded pseudoconvex domain in C 2 {{\mathbf {C}}^2} is a domain of holomorphy.

Polynomial hulls with convex fibers

Polynomial hulls with convex fibers

A Note on Projective Capacity

Introduction. In [1] we defined a capacity in Cn. Recently Molzon, Shiffman and Sibony [8] have introduced a different capacity which is useful for certain Bezout estimates. The object of this note is to apply the methods of [1] to study the capacity of [8]. We shall obtain an equivalent definition of this capacity via Tchebycheff polynomials, along the lines of [1]. Half of this equivalence was independently obtained by Sibony [9].To establish the full equivalence of these two approaches to capacity a notion of Jensen measures in a setting more general than uniform algebras is needed. We shall consider Jensen measures for multiplicative semigroups; these are sets of functions in which only the multiplicative structure is postulated. It will also be useful to generalize the notion of polynomial hull in Cn to a hull with respect to a multiplicative semigroup of polynomials. We can then adapt the approach of [1] to these semigroups.

Complementary Components of Polynomial Hulls

Complementary Components of Polynomial Hulls

Complementary components of polynomial hulls

Let X be a compact subset of the unit sphere S in C n , n &gt; 1 {{\mathbf {C}}^n},n &gt; 1 . It is shown that if a point z is not in the polynomially convex hull of X, then there is a complementary component U of X in S such that z is not in the hull of S U.

Roll Damping Moment of Numerical Models in Forward Motion

In this paper the forward velocity dependence of the roll damping moment on a ship is treated both experimentally and theoretically. Forced rolling tests using polynomial hull form models were conducted to measure the roll damping moment in forward motion. Influences of forward velocity, rolling frequency and principal dimensions are investigated. A theoretical prediction method, which is based upon the thin ship theory and modified to include effects of the ship width approximately, is presented. Calculated values agree well with the experiments through the velocity range in lower frequencies in spite of poor agreements in higher frequency range.

Structure of certain polynomial hulls.

Structure of certain polynomial hulls.