POLYNOMIAL CONVEXITY OF COMPACTS THAT LIES IN CERTAIN LEVI-FLAT HYPERSURFACES IN $\mathbb {C}^2$

A compact subset Y of C n is called polynomially convex if for each point x in C n ~ Y there is a polynomial p such that $$ \left| {p(x)} \right| > \sup \,\left\{ {\left| {p(y)} \right|:y \in Y} \right\} $$ we say that such a polynomial p separates x from Y. A set Y is convex in the ordinary sense just in case each point of the complement can be separated from Y by a polynomial of degree one, so each convex set is polynomially convex. Finite sets are also polynomially convex, so polynomially convex sets need not be connected. Any compact set which lies entirely in the real subspace of points having all real coordinates is polynomially convex because of the Weierstrass approximation theorem. Polynomial convexity, unlike ordinary convexity, is not preserved under real linear transformations or even under general complex linear mappings, though of course it is preserved by complex linear isomorphisms. There is no nice internal description of polynomially convex sets analogous to the condition that a set is convex if and only if it contains the closed line segment joining each pair of its points. Thus although many sets are known to be polynomially convex it can be rather difficult to decide about some fairly simple sets.

In this paper, we first construct a new generalization of $n$-polynomial convex function. That is, this study is a generalization of the definition of "$n$-polynomial convexity" previously found in the literature. By making use of this construction, we derive certain inequalities for this new generalization and show that the first derivative in absolute value corresponds to a new class of $n$-polynomial convexity. Also, we see that the obtained results in the paper while comparing with Hölder, Hölder-İşcan and power-mean, improved-power-mean integral inequalities show that the results give a better approach than the others. Finally, we conclude our paper with applications containing some means.

We remark that as a consequence of Taylor’s Theorem, polynomially convex sets are the compact sets that are holomorphically convex relative to the ambient space C. The problem of deciding whether a compact set in C is polynomially convex or not is a fundamental and difficult problem in complex analysis. We note that polynomial convexity is a global condition. A closed subset E ⊂ C is called locally polynomially convex at z ∈ E if there exists r > 0 such that E ∩ clos B(z, r) is polynomially convex, where

In this article we study generic properties of totally real submanifolds M of . On the one hand we show that if M is polynomially convex and also has bounded exhaustion hulls, then sufficiently small -perturbations of M are polynomially convex and have bounded exhaustion hulls. The proof of this relies on an explicit local function (the Gaussian kernel) and the solution of a Cousin I problem. On the other hand, if , there exist arbitrarily small -perturbations M′ of M such that M′ is polynomially convex and has bounded exhaustion hulls. The proof is a further development of an idea of Forstnerič and Rosay to push different pieces of M into different complex submanifolds of . We also give a perturbation result that can be obtained by applying a holomorphic automorphism.

In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in \(\bigcup _{\theta \in [0,\pi /2]}e^{i\theta }V\) is polynomially convex, where V is a Lagrangian subspace of \(\mathbb {C}^n\). (ii) We show that any compact subset K of \(\{(z,w)\in \mathbb {C}^2:q(w)=\overline{p(z)}\}\), where p and q are two non-constant holomorphic polynomials in one variable, is polynomially convex and \(\mathcal {P}(K)=\mathcal {C}(K)\). KeywordsPolynomial convexityClosed ballTotally realLagrangians2010 Mathematics Subject ClassificationPrimary: 32E20

In this paper, we first consider the graph of $$(F_1,F_{2},\ldots ,F_{n})$$ on $$\overline{\mathbb {D}}^{n},$$ where $$F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\ldots ,n,$$ which has non-isolated CR-singularities if $$m_{j}>1$$ for some $$j\in \{1,2,\ldots ,n\}.$$ We show that under the certain condition on $$R_{j},$$ the graph is polynomially convex, and holomorphic polynomials on the graph approximate all continuous functions. We also show that there exists an open polydisk D centered at the origin such that the set $$\{(z^{m_{1}}_{1},\ldots , z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1}(z),\ldots , \bar{z_{n}}^{m_{2n}} + R_{n}(z)):z\in \overline{D},m_{j}\in \mathbb {N}, j=1,\ldots ,2n\}$$ is polynomially convex, and if $$\gcd (m_{j},m_{k})=1~~\forall j\not =k,$$ the algebra generated by the functions $$z^{m_{1}}_{1},\ldots , z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1},\ldots , \bar{z_{n}}^{m_{2n}} + R_{n}$$ is dense in $$\mathcal {C}(\overline{D}).$$ We prove an analogue of Minsker’s theorem over the closed unit polydisk, i.e., if $$\gcd (m_{j},m_{k})=1~~\forall j\not =k,$$ the algebra $$[z^{m_{1}}_{1},\ldots , z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}},\ldots , \bar{z_{n}}^{m_{2n}};{\overline{\mathbb {D}}^{n}} ]=\mathcal {C}(\overline{\mathbb {D}}^{n}).$$

We study normal forms of germs of singular real-analytic Levi-flat hypersurfaces. We prove the existence of rigid normal forms for singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex quasi-homogeneous polynomials with isolated singularity. This result generalizes previous results of Burns-Gong and Fern\'andez-P\'erez. Furthermore, we prove the existence of two new rigid normal forms for singular real-analytic Levi-flat hypersurfaces which are preserved by a change of isochore coordinates, that is, a change of coordinates that preserves volume.

We show that certain compact polynomially convex subsets of C n {\mathbb {C}^n} remain polynomially convex under sufficiently small C 2 {{\mathbf {C}}^2} perturbations.

Polynomial convexity and strong disk property

The main properties of polynomially convex sets discussed in this chapter are of a general character in that they do not depend on particular structural properties of the sets involved. Section 2.1 contains some of the information about polynomially convex sets that can be derived from the theory of the Cousin problems. Section 2.2 contains two characterizations of polynomially convex sets. Section 2.3 brings the geometric methods of Morse theory and algebraic topology to bear on polynomial convexity. Section 2.4 is devoted to some results for various classes of compacta in Stein manifolds that are parallel to results for polynomially convex subsets of ℂN.KeywordsComplex ManifoldMorse TheoryPseudoconvex DomainMorse FunctionContinuous FamilyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce different notions of polynomial convexity with bounds on degrees of polynomials in C n . We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree bounds for certain sets of points in C and for certain arcs in the unit circle.

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

This paper is concerned with algebraic relaxations, based on the concept of sum of squares decomposition, that give sufficient conditions for convexity of polynomials and can be checked efficiently with semidefinite programming. We propose three natural sum of squares relaxations for polynomial convexity based on respectively, the definition of convexity, the first order characterization of convexity, and the second order characterization of convexity. The main contribution of the paper is to show that all three formulations are equivalent; (though the condition based on the second order characterization leads to semidefinite programs that can be solved much more efficiently). This equivalence result serves as a direct algebraic analogue of a classical result in convex analysis. We also discuss recent related work in the control literature that introduces different sum of squares relaxations for polynomial convexity. We show that the relaxations proposed in our paper will always do at least as well the ones introduced in that work, with significantly less computational effort. Finally, we show that contrary to a claim made in the same related work, if an even degree polynomial is homogeneous, then it is quasiconvex if and only if it is convex. An example is given.

We use our disc formula for the Siciak–Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs. The polynomial hull K of a compact subset K of complex affine space C n is the compact set of those x ∈ C for which |P (x)| ≤ supK |P | for all complex polynomials P in n variables. The set K is said to be polynomially convex if K = K. Polynomial hulls are usually difficult to determine. By the maximum principle for holomorphic functions it is clear that x ∈ K if there is a continuous map f from the closed unit disc D into C, holomorphic on D, such that f(0) = x and f maps the unit circle into K. In the early days of polynomial convexity theory it seemed possible that K might simply be the union of all analytic discs in C with boundary in K. Stolzenberg’s counterexample of 1963 [7] showed that K \K may be nonempty—and in fact quite large (see [1])—without containing any nonconstant analytic discs. The question of whether polynomial hulls could nevertheless be somehow described in terms of analytic discs remained open for three decades, until Poletsky derived an answer from his theory of disc functionals [6] (see Theorem 2 below). In this note, we give a different characterization of the polynomial hull of a connected compact subset of C, based on our generalization in [5] of Lempert’s disc formula for the Siciak–Zahariuta extremal function from the convex case to the connected case. The gist of both results, Poletsky’s and ours, is to suitably weaken the requirement that the analytic discs in the 2000 Mathematics Subject Classification: Primary 32E20; Secondary 32U35.

A note on polynomial convexity of the union of finitely many totally-real planes in [formula omitted