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arXiv · 2610.16724

Universal Varieties for Signatures of Membranes

Carlos Améndola·Gabriel Riffo·Leonard Schmitz
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Abstract

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

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Carlos Améndola, Gabriel Riffo, Leonard Schmitz. 2026-10-12. Universal Varieties for Signatures of Membranes. https://arxiv.org/abs/2610.16724

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