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 ℝ n × ℝ (space × time), i.e., polynomial solutions of the heat equation satisfying ∂ t p ¬ ≡ 0 and p λ x , λ 2 t = λ d p ( x , t ) for all x ∈ ℝ n , t ∈ ℝ , and λ > 0 . When n = 1 , it is classically known that the number of nodal domains is precisely 2 ⌈ d / 2 ⌉ . When n = 2 , we prove that the minimum number of nodal domains is 2 if d ¬ ≡ 0 ( mod 4 ) and is 3 if d ≡ 0 ( mod 4 ) . When n ≥ 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 → ∞ and lies between ⌊ d n ⌋ n and 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.