Discussion Overview
The discussion revolves around the nature of analytic functions and their relationship to polynomials, particularly whether an analytic function that can be locally expressed as a power series qualifies as an element of the algebra of polynomials in z. The conversation touches on theoretical aspects and implications for invariant theory and statistical physics.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants propose that an analytic function f(z) can be considered an element of the algebra of polynomials in z if it can be locally expanded in a power series, provided that the power series is finite.
- Others argue that by definition, a polynomial must have a finite highest power, which implies that infinite power series do not qualify as polynomials.
- A later reply expresses interest in extending the concept of polynomials to analytic functions, particularly in the context of invariant theory and statistical physics, and seeks insights on how the "pull back" mechanism might apply to analytic functions.
Areas of Agreement / Disagreement
Participants generally agree on the definition of polynomials as functions with a finite highest power. However, the broader question of whether analytic functions can be classified similarly remains unresolved, with differing views on the implications of power series expansions.
Contextual Notes
The discussion does not resolve the conditions under which an analytic function may or may not be considered a polynomial, nor does it clarify the assumptions regarding the nature of the power series involved.