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arXiv · 2605.18699

Nested nodal loops of biharmonic functions

Javier Gómez-Serrano·Robert Koirala·Alexander Logunov
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Abstract

Given any \(n\in\mathbb{N}\), we construct a real-valued biharmonic polynomial on \(\mathbb{R}^2\) whose zero set contains a nest of \(n\) smooth, disjoint topological loops, meaning that the \(k\)-th loop lies inside the domain bounded by the \((k+1)\)-st loop for \(k=1,\ldots,n-1\). The case \(n=2\), i.e., the existence of two nested loops, is related to the failure of the Boggio-Hadamard conjecture from the early 1900s.

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Javier Gómez-Serrano, Robert Koirala, Alexander Logunov. 2026-05-18. Nested nodal loops of biharmonic functions. https://arxiv.org/abs/2605.18699

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