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Polynomial convexity with degree bounds

Marko Slapar

Abstract

We introduce different notions of polynomial convexity with bounds on degrees of polynomials in $\mathbb C^n$. 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 $\mathbb C$ and for certain arcs in the unit circle.

Polynomial convexity with degree bounds

Abstract

We introduce different notions of polynomial convexity with bounds on degrees of polynomials in . 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 and for certain arcs in the unit circle.
Paper Structure (4 sections, 9 theorems, 51 equations, 3 figures)

This paper contains 4 sections, 9 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results in $\mathbb{C}^2$
  3. Points in $\mathbb{C}$
  4. Subsets on the unit circle

Key Result

Proposition 1

A compact set $K\subset\mathbb{C}^n$ is polynomially convex, if and only if, for every $z_0\in\mathbb{C}^n\backslash K$, there exists a curve of algebraic surfaces $H_t$, $t\ge s$, that tends to infinity, so that $z_0\in H_s$ and $H_t\cap K=\emptyset$ for every $t\ge s$.

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (27)

  • Proposition 1: St, &1
  • Definition 2
  • Remark 3
  • Remark 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Example 7
  • Lemma 8
  • ...and 17 more