On the number of nodal domains of homogeneous caloric polynomials

TL;DR

The paper determines the exact minimal and asymptotic maximal numbers of nodal domains for time-dependent homogeneous caloric polynomials (hcps) in space–time $\mathbb{R}^{n+1}$. By building a basis from basic 1+1D hcps and applying Lewy-type perturbations, it shows $m_{n,d}=2$ for all $n\ge 3$ and $d\ge 2$, while in 2+1 dimensions the minimum depends on $d$ (specifically $m_{2,d}=2$ when $d\not\equiv 0\pmod 4$ and $m_{2,d}=3$ when $d\equiv 0\pmod 4$); it also constructs explicit configurations realizing these bounds. For the maximum, it proves $M_{n,d}=\Theta(d^n)$ as $d\to\infty$, with concrete lower and upper bounds $\Big\lfloor\frac{d}{n}\Big\rfloor^n \le M_{n,d} \le \binom{n+d}{n}$, using product constructions and Courant-type nodal estimates on negative time-slices. An application to free boundary regularity for caloric measure confirms the existence of singular strata in the two-phase setting described by Mourgoglou and Puliatti. Overall, the work clarifies the nodal-topology landscape of parabolic polynomials and provides tools for understanding the geometry of caloric measure in singular regimes.

Abstract

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 $\partial_t p\not\equiv 0$ and $$p(λx, λ^2 t) = λ^d p(x,t)\quad\text{for all $x \in \mathbb{R}^n$, $t \in \mathbb{R}$, and $λ> 0$.}$$ When $n=1$, it is classically known that the number of nodal domains is precisely $2\lceil d/2\rceil$. When $n=2$, we prove that the minimum number of nodal domains is 2 if $d\not \equiv 0\pmod 4$ and is 3 if $d\equiv 0\pmod 4$. When $n\geq 3$, 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 $Θ(d^n)$ as $d\rightarrow\infty$ and lies between $\lfloor \frac{d}{n}\rfloor^n$ and $\binom{n+d}{n}$ 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.

Figures (12)

• Figure 1.1: Gallery of nodal sets of homogeneous caloric polynomials $u$ in 2+1 dimensions achieving the minimum number $m_{2,d}$ of nodal domains. For increased visibility, we show the intersection of the full nodal set with an annulus: $\{ (x,y,t) \, : \, u(x,y,t) = 0 \} \cap (B_1 \setminus B_{1-\delta})$. Left: \ref{['eqn:u_0_mod_4']} with $d=4$, $\varepsilon = 0.2$, $\alpha = \pi/10$. Middle: \ref{['eqn:d_odd_u']} with $d = 5$, $\varepsilon =0.3$, $\alpha = \pi/10$. Right: \ref{['eqn:lewy_u']} with $d = 6$, $\varepsilon = 0.05$.
• Figure 2.1: The nodal set of a degree $d$ hcp in $\mathbb{R}^{1+1}$ is a union of $\lfloor d/2 \rfloor$ nested, downward-opening parabolas with a common turning point at the origin, and when $d$ is odd, an additional vertical line (the $t$-axis). Thus, the number of nodal domains is precisely $2\lceil d/2\rceil$. From left to right, we illustrate the cases $d=2$, …, $d=5$. Inside the nodal set of $p_dp_{d+1}$, the "nodal parabolas" of consecutive hcps $p_d$ and $p_{d+1}$ are intertwined: the "widest" parabola of $p_{d+1}$ sits above the "widest" parabola of $p_d$; the "widest" parabola of $p_d$ sits above the "second widest" parabola of $p_{d+1}$; etc.
• Figure 3.1: Connecting $(v_1,w_1)=(x_1,y_1,z_1,t_1)$ to $(v_2,w_2)=(x_2,y_2,z_2,t_2)$ in $\mathbb{R}^{3+1}\cap\{u>0\}$, where $\phi(x,y)$ is a degree 4 hhp in $\mathbb{R}^{2}$ (left) and $\psi(z,t)$ is a degree 4 hcp in $\mathbb{R}^{1+1}$ (right).
• Figure 4.1: Parameters $n^+_{\mathrm{in}}$ and $n^+_{\mathrm{out}}$ count the number of positive chambers of $u$ near the origin and near infinity, respectively. On the right, we illustrate the base case $n^+_{\mathrm{in}} = 2, n^+_{\mathrm{out}} =1$ (with zero negative chambers at infinity).
• Figure 4.2: Proof of Proposition \ref{['prop:low_bd_2']}. If the union $\tilde{\gamma}$ of the path $\gamma$ and its reflection $R\circ \gamma$ lie in the positivity set of $p|_{\mathbb{S}^2}$, then the negativity set of $p|_{\mathbb{S}^2}$ is disconnected.

Theorems & Definitions (65)

• Theorem 1.1: minimum number of nodal domains
• Theorem 1.2: maximum number of nodal domains
• Remark 1.3
• Definition 1.4
• Definition 1.5
• Example 1.6
• Remark 1.7
• Definition 1.8
• Remark 1.9
• Remark 1.10