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Signature matrices
of membranes
Felix Lotter and Leonard Schmitz |
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Vol. 17 (2026), No. 1, 75–99
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Abstract |
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The signature of a membrane is a sequence of tensors whose entries are iterated integrals. We study algebraic properties of membrane signatures, with a focus on signature matrices of polynomial and piecewise bilinear membranes. Generalizing known results for path signatures, we show that the two families of membranes admit the same set of signature matrices and we examine the corresponding affine varieties. In particular, we prove that there are no algebraic relations on signature matrices of membranes, in contrast to the path case. We complement our results by a linear time algorithm for the computation of signature tensors for piecewise bilinear membranes. |
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Keywords
two-parameter signatures, affine algebraic varieties,
matrix congruence, geometry of paths and membranes,
iterated integrals
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Mathematical Subject Classification
Primary: 13P25, 14Q15, 60L10
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Milestones
Received: 11 June 2025
Revised: 4 December 2025
Accepted: 20 December 2025
Published: 15 February 2026
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Authors |
| Felix Lotter | |
| Max Planck Institute for Mathematics
in the Sciences Leipzig Germany |
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| Leonard Schmitz | |
| Technische Universität Berlin Berlin Germany |
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| © 2026 MSP (Mathematical Sciences Publishers). |
