On the Holomorphicity of Hartogs Series Satisfying Algebraic Relations: Necessary Conditions and Counterexamples

Core Concepts

While the convergence of a Hartogs series at points in a dense subset of a domain is a known necessary condition for its holomorphicity, algebraicity of the series over the ring of holomorphic functions is also crucial, as demonstrated by a counterexample involving the elliptic theta function.

Abstract

Bibliographic Information:

Aoki, H., & Saito, K. (2024). On holomorphicity of Hartogs series satisfying algebraic relations. arXiv preprint arXiv:2411.10641v1.

Research Objective:

This research paper investigates the conditions under which a Hartogs series, a formal power series with holomorphic coefficients, defines a holomorphic function on a given domain. Specifically, it examines the role of algebraicity as a necessary condition for holomorphicity.

Methodology:

The authors employ techniques from complex analysis, particularly focusing on properties of holomorphic functions, analytic sets, and the Cauchy-Hadamard theorem for convergence of power series. They prove their main theorem through a four-step process involving the analysis of the discriminant of the polynomial relation satisfied by the Hartogs series.

Key Findings:

The paper proves that if a Hartogs series converges on a dense subset of its domain and is algebraic over the ring of holomorphic functions on that domain, then the series converges on the entire domain and defines a holomorphic function. The authors further provide a counterexample using the elliptic theta function, demonstrating that algebraicity is an essential condition for this result to hold.

Main Conclusions:

The study establishes algebraicity as a critical factor in determining the holomorphicity of Hartogs series. This finding has implications for the study of automorphic forms, particularly in the context of Fourier-Jacobi series expansions.

Significance:

This research contributes to the understanding of Hartogs series and their convergence properties, with potential applications in complex analysis and related fields like number theory and mathematical physics.

Limitations and Future Research:

The paper primarily focuses on domains in CN. Exploring similar results for more general complex manifolds could be an area for future research. Additionally, investigating the interplay between the density of the convergence set and the degree of the algebraic relation could yield further insights.

Stats

Quotes

"The aim of the present note is to prepare a general theorem which can be applied to the above problem."
"The condition (C2) is essentially necessary in Theorem 1."

Key Insights Distilled From

On holomorphicity of Hartogs series satisfying algebraic relations

by Hiroki Aoki,... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.10641.pdf

Deeper Inquiries

How can the findings of this paper be extended to study the convergence of other types of series expansions, such as those arising in different areas of mathematical physics?

This paper focuses on Hartogs series, which are power series in one complex variable whose coefficients are holomorphic functions in other variables. The key result is that such series, if algebraic over a ring of holomorphic functions and convergent on a dense subset, are actually holomorphic everywhere. This idea of leveraging algebraicity and dense convergence could potentially be extended to other series expansions appearing in mathematical physics. Here's how:

Identifying Analogous Structures: The first step would be to identify series expansions in mathematical physics that share structural similarities with Hartogs series. This means looking for expansions where:

The series coefficients are themselves functions with "nice" properties (like holomorphicity).
There's a notion of algebraicity over a suitable ring of functions relevant to the physical problem.
Dense convergence might arise naturally from physical considerations or approximations.

Formalizing Algebraicity: Once a suitable series expansion is identified, the concept of algebraicity needs to be carefully defined in that context. This might involve:

Identifying the appropriate ring of functions over which algebraicity is considered. This ring should capture the essential properties of the functions appearing as coefficients.
Determining what it means for the series to satisfy an algebraic relation over this ring. This might involve differential equations, recurrence relations, or other constraints.

Adapting the Proof Strategy: The proof strategy used in the paper for Hartogs series relies on tools from complex analysis, particularly the properties of holomorphic functions and analytic sets. Adapting this strategy would require:

Finding analogous tools and techniques in the relevant area of mathematical physics. For example, if dealing with series solutions to differential equations, one might use tools from functional analysis or the theory of differential equations.
Carefully modifying the arguments to account for the specific properties of the series expansion and the underlying mathematical framework.

Examples in Mathematical Physics:

Perturbation Series: In quantum field theory, perturbation series are used to approximate solutions. These series often have coefficients that are themselves functions of physical parameters. Investigating algebraicity conditions for convergence could provide insights into the range of validity of these approximations.
Fourier Series Solutions: In many physical problems, solutions are sought in the form of Fourier series. The coefficients in these series are often determined by the initial conditions or boundary conditions of the problem. Exploring algebraicity in this context could lead to new criteria for the convergence of these solutions.

Could there be alternative conditions, besides algebraicity, that ensure the holomorphicity of a Hartogs series converging on a dense subset?

Yes, while algebraicity provides a sufficient condition for holomorphicity in the context of this paper, alternative conditions could exist. Here are some possibilities:

Growth Conditions: Instead of algebraicity, one could impose restrictions on the growth rate of the coefficients f_n(z) as n tends to infinity. For example:

Bounded Growth: If the coefficients are uniformly bounded on compact subsets of the domain, the series might converge to a holomorphic function.
Moderate Growth: Conditions like |f_n(z)| ≤ C(z) n^k for some positive constant k and a function C(z) bounded on compact sets could be explored.

Regularity Conditions: Requiring additional regularity properties of the coefficients beyond holomorphicity might lead to holomorphicity of the series. Examples include:

Real Analyticity: If the coefficients are real analytic, stronger convergence results might hold.
Smoothness: Even requiring the coefficients to be infinitely differentiable (smooth) could potentially lead to different sufficient conditions.

Geometric Conditions: The geometry of the domain where the series converges densely could also play a role. For instance:

Pseudoconvexity: If the domain has a property called pseudoconvexity, it might impose stronger constraints on the behavior of holomorphic functions, potentially leading to holomorphicity of the series under weaker assumptions.

Approximation Properties: Conditions related to how well the series can be approximated by holomorphic functions could be considered. For example:

Uniform Approximation on Compacts: If the series can be uniformly approximated on compact subsets by holomorphic functions, it might itself converge to a holomorphic function.

Exploring these alternative conditions would require careful analysis using tools from complex analysis, functional analysis, and potentially other areas of mathematics.

What are the implications of this research for the study of complex manifolds and their geometric properties?

This research, while seemingly focused on the convergence of a specific type of series, has potential implications for the study of complex manifolds and their geometry:

Characterizing Holomorphic Functions: The result provides a new way to characterize holomorphic functions on complex manifolds. Instead of directly verifying the Cauchy-Riemann equations, one could potentially construct a function as a Hartogs series and then check for algebraicity and dense convergence. This could be particularly useful in cases where directly verifying holomorphicity is difficult.

Understanding Analytic Subsets: The proof of the main theorem relies heavily on the properties of analytic subsets. The fact that algebraicity and dense convergence of a Hartogs series imply holomorphicity provides an indirect link between the algebraic properties of the series and the geometric properties of the domain, particularly its analytic subsets.

Constructing Holomorphic Objects: The result could potentially be used to construct holomorphic objects on complex manifolds. For example, one might be able to construct holomorphic sections of vector bundles or holomorphic maps between complex manifolds by representing them as Hartogs series and then verifying the required conditions.

Connections to Automorphic Forms: The authors mention that their motivation stems from the theory of automorphic forms, which are functions on complex manifolds with certain transformation properties. The result might provide new tools for studying the convergence and properties of series expansions arising in the theory of automorphic forms, potentially leading to a deeper understanding of these important objects.

Generalizations and Extensions: The ideas presented in the paper could inspire further research into generalizations and extensions of the main result. For example, one could investigate:

Analogous results for other types of series expansions relevant to complex geometry.
The implications of the result for the study of singularities of holomorphic functions and complex manifolds.
Connections to other areas of complex geometry, such as Kähler geometry and Hodge theory.

Overall, this research highlights a fruitful interplay between complex analysis, algebra, and geometry. It suggests that seemingly technical results about the convergence of series can have deeper implications for our understanding of complex manifolds and their properties.