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Boundary local integrability of rational functions in two variables

Greg Knese

TL;DR

This work provides a complete local theory for the boundary integrability of rational functions in two variables, focusing on quotients $Q/P$ with $P$ nonvanishing in the bi-upper half-plane but vanishing at the origin. By replacing $P$ with a robust local model $[P]$ and exploiting a two-variable branch-matching framework via the auxiliary polynomial $A+tB$, the authors derive explicit necessary-and-sufficient criteria for $Q/P$ to belong to $L^{\mathfrak{p}}_{loc}$, parameterized by Puiseux-type data $L_j$ and $q_j$. They give precise dimension counts for the relevant local ideals and furnish a constructive basis for the quotient $\mathbb{C}\{x,y\}/(P,\bar P)$, enabling concrete test criteria and computable invariants. Applications include a result that every bounded rational function on the bidisk has derivatives on the torus in $L^1$, and a new, self-contained proof of Kollár’s description of the integrally closed ideal associated to $P$. The approach, rooted in a local model for stable polynomials and refined one- and two-variable analyses, offers a flexible toolkit for addressing other local questions about stable polynomials.

Abstract

Motivated by studying boundary singularities of rational functions in two variables that are analytic on a domain, we investigate local integrability on $\mathbb{R}^2$ near $(0,0)$ of rational functions with denominator non-vanishing in the bi-upper half-plane but with an isolated zero (with respect to $\mathbb{R}^2$) at the origin. Building on work of Bickel-Pascoe-Sola, we give a necessary and sufficient test for membership in a local $L^{p}(\mathbb{R}^2)$ space and we give a complete description of all numerators $Q$ such that $Q/P$ is locally in a given $L^{p}$ space. As applications, we prove that every bounded rational function on the bidisk has partial derivatives belonging to $L^1$ on the two-torus. In addition, we give a new proof of a conjecture, started in Bickel-Knese-Pascoe-Sola and completed by Kollár, characterizing the ideal of $Q$ such that $Q/P$ is locally bounded. A larger takeaway from this work is that a local model for stable polynomials we employ is a flexible tool and may be of use for other local questions about stable polynomials.

Boundary local integrability of rational functions in two variables

TL;DR

This work provides a complete local theory for the boundary integrability of rational functions in two variables, focusing on quotients with nonvanishing in the bi-upper half-plane but vanishing at the origin. By replacing with a robust local model and exploiting a two-variable branch-matching framework via the auxiliary polynomial , the authors derive explicit necessary-and-sufficient criteria for to belong to , parameterized by Puiseux-type data and . They give precise dimension counts for the relevant local ideals and furnish a constructive basis for the quotient , enabling concrete test criteria and computable invariants. Applications include a result that every bounded rational function on the bidisk has derivatives on the torus in , and a new, self-contained proof of Kollár’s description of the integrally closed ideal associated to . The approach, rooted in a local model for stable polynomials and refined one- and two-variable analyses, offers a flexible toolkit for addressing other local questions about stable polynomials.

Abstract

Motivated by studying boundary singularities of rational functions in two variables that are analytic on a domain, we investigate local integrability on near of rational functions with denominator non-vanishing in the bi-upper half-plane but with an isolated zero (with respect to ) at the origin. Building on work of Bickel-Pascoe-Sola, we give a necessary and sufficient test for membership in a local space and we give a complete description of all numerators such that is locally in a given space. As applications, we prove that every bounded rational function on the bidisk has partial derivatives belonging to on the two-torus. In addition, we give a new proof of a conjecture, started in Bickel-Knese-Pascoe-Sola and completed by Kollár, characterizing the ideal of such that is locally bounded. A larger takeaway from this work is that a local model for stable polynomials we employ is a flexible tool and may be of use for other local questions about stable polynomials.
Paper Structure (19 sections, 49 theorems, 359 equations)

This paper contains 19 sections, 49 theorems, 359 equations.

Table of Contents

  1. Introduction
  2. First Example
  3. Main Theorems
  4. Outline of our approach
  5. Local theory for polynomials with no zeros on $\mathbb{R}\times \mathbb{H}$
  6. Matching Branches
  7. One variable facts and formulas
  8. $L^{\mathfrak{p}}$-theory in one variable
  9. Characterization of local integrability
  10. Characterization of local $L^{\mathfrak{p}}$ integrability
  11. Final integrability characterization---Theorem \ref{['finalint']}
  12. Derivative integrability
  13. A basis for $\mathbb{C}\{x,y\}/(P,\bar{P})$
  14. Characterization of integrability quotients
  15. Working out the multiplicity three case
  16. ...and 4 more sections

Key Result

theorem 1.2

[Theorem 1.2 BKPS] Let $P(x,y) \in \mathbb{C}[x,y]$ have no zeros in $\mathbb{H}^2$ and no common factors with $\bar{P}$. Suppose $P$ vanishes to order $M$ at $(0,0)$. Then there exist natural numbers $L_1,\dots, L_M \geq 1$ and real coefficient polynomials $q_1(x),\dots, q_M(x) \in \mathbb{R}[x]$ s for $j=1,\dots, M$ such that if we define then are bounded in a punctured neighborhood of $(0,0)$

Theorems & Definitions (98)

  • Example 1.1
  • theorem 1.2
  • Definition 1.2
  • Definition 1.2
  • Example 1.3
  • Definition 1.3
  • theorem 1.4
  • Example 1.5
  • theorem 1.6
  • theorem 1.7
  • ...and 88 more