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

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 $\gamma=\frac{2m+3}{2m+1}$.