On the number of nodal domains of homogeneous caloric polynomials

We investigate the minimum and maximum number of nodal domains across all time-dependent homogeneous caloric polynomials of degree $d$ in ${\mathbb{R}}^n\times \mathbb{R}$ (space $\times$ time), i.e., polynomial solutions of the heat equation satisfying ${\mathrm{\partial }}_{t}p\equiv ̸0$ and $$p(\lambda x,{\lambda }^{2}t)={\lambda }^{d}p(x,t)\text{for all }x\in {\mathbb{R}}^n\text{, }t\in \mathbb{R}\text{, and }\lambda >0\text{.}$$ When $n=1$, it is classically known that the number of nodal domains is precisely $2\lceil d\mathrm{/}2\rceil$. When $n=2$, we prove that the minimum number of nodal domains is 2 if $d\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and is 3 if $d\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$. When $n\ge 3$, we prove that the minimum number of nodal domains is $2$ for all $d$. Finally, we show that the maximum number of nodal domains is $\mathrm{\Theta }(d^n)$ as $d\to \mathrm{\infty }$ and lies between $\lfloor \frac{d}{n}{\rfloor }^n$ and $\left(\genfrac{}{}{0px}{}{n+d}{n}\right)$ for all $n$ and $d$. As an application and motivation for counting nodal domains, we confirm existence of the singular strata in Mourgoglou and Puliatti’s two-phase free boundary regularity theorem for caloric measure.