Monomial Moving Zeros for the Heat Semigroup: An Explicit Finite-Order Differential Hierarchy
Abstract
Let Pt=exp(tD2/2) be the heat semigroup. For every integer k≥2 and parameter ρ∈C, we construct an entire function Γρ,k whose heat evolution has the prescribed moving zero x=ρtk and constant spatial flux there: (PtΓρ,k)(ρtk)=0, ∂x(PtΓρ,k)(ρtk)=1. The initial function is explicit, Γρ,k(x) =x∑q≥0((−2kρx2k−1)q)/q! (kq)!/(((2k−1)q+1)!), and is a generalized hypergeometric function whose entire-function order is strictly below two. Most importantly, it is annihilated by an explicit polynomial-coefficient differential operator of order 2k−1. Thus every monomial moving boundary admits a finite-order stationary realization after a natural boundary-flux selection, even though the naive order-k family Dk+a xk−1 already fails to produce the exact boundary at k=4. No minimality claim is made: the result gives a universal finite-order hierarchy, while the least possible differential order remains open and depends on the choice made inside the inverse kernel.
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BibTeX
@misc{hernandezdelvalle2026monomial,
title = {{Monomial Moving Zeros for the Heat Semigroup: An Explicit Finite-Order Differential Hierarchy}},
author = {Gerardo Hernandez-del-Valle},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27326},
eprint = {2610.27326},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27326},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Monomial Moving Zeros for the Heat Semigroup: An Explicit Finite-Order Differential Hierarchy
AU - Gerardo Hernandez-del-Valle
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27326
UR - https://arxiv.org/abs/2610.27326
AB - Let P_t=exp(tD^2/2) be the heat semigroup. For every integer k≥2 and parameter ρ∈C, we construct an entire function Γ_ρ,k whose heat evolution has the prescribed moving zero x=ρt^k and constant spatial flux there: (P_tΓ_ρ,k)(ρt^k)=0, ∂_x(P_tΓ_ρ,k)(ρt^k)=1. The initial function is explicit, Γ_ρ,k(x) =x∑_q≥0((−2^kρx^2k−1)^q)/q! (kq)!/(((2k−1)q+1)!), and is a generalized hypergeometric function whose entire-function order is strictly below two. Most importantly, it is annihilated by an explicit polynomial-coefficient differential operator of order 2k−1. Thus every monomial moving boundary admits a finite-order stationary realization after a natural boundary-flux selection, even though the naive order-k family D^k+a x^k−1 already fails to produce the exact boundary at k=4. No minimality claim is made: the result gives a universal finite-order hierarchy, while the least possible differential order remains open and depends on the choice made inside the inverse kernel.
ER -