Algebraic structure of quasiradial solutions to the $\gamma$-harmonic equation

TL;DR

This paper provides an explicit algebraic representation of quasiradial $ ext{ extgamma}$-harmonic functions, classifies the values of $ ext{ extgamma}$ that admit such solutions, and reveals a finite set of solutions for $| ext{ extgamma}|>1$ with a special connection to gas adiabatic constants.

Contribution

It introduces an explicit algebraic form for quasiradial $ ext{ extgamma}$-harmonic functions and characterizes all $ ext{ extgamma}$ values with algebraic solutions, highlighting a finite solution set for $| ext{ extgamma}|>1$.

Findings

Only finitely many algebraic solutions exist for fixed $| ext{ extgamma}|>1$.

Complete description of $ ext{ extgamma}$ values with algebraic quasiradial solutions.

Connection between specific $ ext{ extgamma}$ values and ideal gas adiabatic constants.

Abstract

We obtain an explicit representation for quasiradial $\gamma$-harmonic functions, which shows that these functions have essentially algebraic nature. In particular, we give a complete description of all $\gamma$ which admit algebraic quasiradial solutions. Unlike the cases $\gamma=\infty$ and $\gamma=1$, only finitely many algebraic solutions is shown to exist for any fixed $|\gamma|>1$. Moreover, there is a special extremal series of $\gamma $ which exactly corresponds to the well-known ideal $m$-atomic gas adiabatic constant $\gamma=\frac{2m+3}{2m+1}$.