Signature Varieties in Mathematics

• Signature varieties are algebraic and geometric loci defined by signature constraints that distinguish and classify diverse mathematical structures.
• They arise in contexts such as universal algebra, tensor analysis, and commutative algebra, employing operadic methods, generating series, and polyhedral volume formulas.
• Applications span path analysis in machine learning, singularity detection, and arithmetic density, offering actionable insights for both theory and computation.

Signature varieties encapsulate a spectrum of mathematical objects and concepts that arise when the notion of “signature” is used to distinguish structures in algebra, geometry, and arithmetic. Across diverse contexts—universal algebras, commutative algebra, path spaces, and number theory—the term “signature variety” denotes either the class of algebras or geometric loci satisfying given signature constraints, or concrete algebraic varieties defined by polynomial relations among signature invariants. This article surveys the principal mathematical frameworks in which signature varieties are defined and studied, and presents the foundational structures, central results, and interdisciplinary connections as developed in the arXiv literature.

1. Signature Varieties in Universal Algebra and Operad Theory

In the context of universal algebra, a signature is a discrete set of operation symbols (typically with specified arities). A variety of algebras—or “signature variety”—is the class of algebras sharing the same signature and identities. The operadic formalism organizes these varieties: each signature Ω generates a free operad $P$, whose components $P(n)$ are the spaces of all multilinear operations of arity $n$.

• Key equivalence: The category of algebras of signature Ω is equivalent to the category of algebras over the operad $P$ generated by Ω (0905.4748).
• Subvarieties and clones: Subvarieties correspond to quotient operads; clones (linear combinations generated by a single operation under composition) correspond to suboperads.

A central problem addressed in (0905.4748) is the Kurosh problem—whether there exist infinite-dimensional algebraic algebras within such varieties. The paper establishes that given any finite signature, there exists a variety whose free algebra contains multilinear elements of arbitrary degree, but every clone generated by a single multilinear element satisfies a nontrivial identity. When the number of binary operations is at least two, each such clone is finitely dimensional.

The operadic approach leverages homological algebra, particularly resolutions and Tor functors, to paper the structure and growth (via generating series)

$P(z) = \sum_{n \geq 1} \frac{\dim P(n)}{n!} z^n$

and applies an operadic version of the Golod–Shafarevich theorem to construct infinite varieties with locally controlled (finite or nilpotent) behavior.

2. Signature Tensor Varieties and Path Geometry

In stochastic analysis, algebraic geometry, and machine learning, signatures denote sequences of tensors encoding iterated integrals along paths:

$$\sigma^{(k)}(X) = \int_{0 \le t_1 \le \cdots \le t_k \le 1} dX(t_1) \otimes \cdots \otimes dX(t_k)$$

The collection of all signature tensors of a path $X:[0,1] \to \mathbb{R}^d$ forms a point in an infinite-dimensional tensor space; when restricted to finite degrees $k$, the image lies in a finite-dimensional projective space.

• The signature tensor variety is the Zariski closure of the set of all signature tensors for a class of paths (piecewise linear, polynomial, or general smooth paths).
• By Chen’s theorem, the set of all step-$n$ signatures coincides with the exponential of the free nilpotent Lie group $\mathcal{G}^n(\mathbb{R}^d)$.
• Signature tensor varieties admit description by shuffle polynomial relations (e.g., $\sigma_{I}\sigma_J = \sigma_{I \shuffle J}$) and form algebraic varieties such as $\mathcal{U}_{d,k}$ (“universal signature variety”), $\mathcal{P}_{d,k,m}$ (polynomial paths), or $\mathcal{L}_{d,k,m}$ (piecewise linear paths) (Améndola et al., 2018).

These varieties are now at the center of efforts to characterize the geometry, identifiability, and analytic properties of path signatures for deterministic and random paths. In the stochastic setting, expected signatures of Brownian motions yield varieties (e.g., $\mathcal{B}_{d,k}$) with remarkable algebraic structure, often captured by exponential formulas (e.g., $$\mathbb{E}(\sigma(X)) = \exp(\mu + \tfrac{1}{2}\Sigma)$$ for drift $\mu$ and covariance $\Sigma$).

3. Discrete Signature Varieties

In time series analysis, discrete signatures are computed from sequences $x: \{1, ..., N\} \to K^d$ by summing over iterated increments:

$$\left\langle \mathcal{S}(x), p_1 \otimes \cdots \otimes p_\ell \right\rangle = \sum_{1 \leq i_1 < \cdots < i_\ell < N} p_1(\Delta x_{i_1}) \cdots p_\ell(\Delta x_{i_\ell})$$

for nonconstant monomials $p_j$, and $\Delta x_i = x_{i+1} - x_i$.

The discrete signature variety is defined as the Zariski closure of the images of these polynomial maps at fixed level $h$, yielding an irreducible variety in projective space $\mathbb{P}^{n-1}$, with $n$ determined by the degree and dimension. These varieties exhibit:

• Time-warping invariance: Repetitions of consecutive data do not affect the discrete signature.
• Universal stabilization: For large $N$, the variety stabilizes, denoted $\mathcal{V}_{d,h}$.
• Quasi-shuffle identities: Discrete signatures satisfy quasi-shuffle, rather than pure shuffle, relations, impacting the algebraic type of their vanishing ideals (Bellingeri et al., 2023).

Key computational results provide explicit formulas for the dimension using Lyndon word counts and Möbius function combinatorics. Partial results on discrete Chen–Chow theorems are established for weight-2 signatures over complex numbers.

4. Algebraic Geometry of Path Signatures

Recent developments generalize the algebraic framework to the space of paths itself, introducing a Zariski topology on the infinite-dimensional space of paths $2^{\mathbb{R}^d}$, with closed sets defined by vanishing shuffle ideals in the tensor algebra $T(\mathbb{R}^d)$ (Preiß, 2023). Essential aspects include:

• Path varieties: Sets of paths whose signatures satisfy specified polynomial equations.
• Halfshuffle structures: Non-associative halfshuffle operations capture stability under subpaths and propagation of polynomial equations.
• Concatenation stability: Path varieties are stable under concatenation provided the defining ideal is a Hopf ideal.
• Generalization of classic varieties: Signature varieties allow the description—via iterated-integral equations—of generalized algebraic sets such as the graph of the exponential function, which is not an algebraic curve in the classical sense.
• Infinite-dimension and reducibility: The notion of (possibly infinite) dimension for path varieties is articulated using countable reducibility and subvariety chains.

5. Signature Varieties in Commutative Algebra (F-Signature)

In positive characteristic commutative algebra, the F-signature is defined for a Noetherian ring $R$ (affine or toric) as

$s(R) = \lim_{e \to \infty} \frac{a_e}{p^{e d}}$

with $a_e$ the number of free $R$-summands in $R^{1/p^e}$ and $d = \dim R$ (Korff, 2011). In the toric context, the F-signature quantifies the asymptotic density of free direct summands, serving as a singularity invariant:

• $s(R) = 1$ if and only if $R$ is regular; $s(R) > 0$ if and only if $R$ is strongly F-regular.
• Volume formula: $s(R)$ is the volume of a polytope $P_\sigma$ determined by the toric data:

$$P_\sigma = \{ w \in M_\mathbb{R} : 0 \leq w \cdot v_i < 1,\ \forall i \}$$

This polyhedral description extends to toric pairs and triples and has computational and theoretical implications for Hilbert–Kunz multiplicities, tight closure theory, and the minimal model program.

6. Signature Varieties in Noncommutative Algebra and Curvature

In free (noncommutative) algebraic geometry, varieties are defined via the zero set $V(p)$ of a symmetric noncommutative polynomial $p$ (Dym et al., 2012). The signature refers to the positive and negative eigenvalue counts of the noncommutative second fundamental form, computed via Hessians on the clamped tangent space:

$$\text{At } (X, v),\ T_p(X, v) = \{ H : p'(X)[H] v = 0 \}$$

$H \mapsto - (p''(X)[H] v, v)$

A sum–difference of squares (SDS) decomposition yields minimal counts $c_+(X, v; p),\ c_-(X, v; p)$, which control geometric invariants—in particular, the degree of $p$:

$\deg(p) \leq 2 C_+(S) + 2$

where $C_+(S)$ is the supremum of $c_+$ over all direct sums of points in the variety.

7. Signature Varieties in Arithmetic Geometry

In arithmetic geometry, signature varieties refer to the collection of abelian varieties (typically with CM by a field $F$) satisfying specific signature constraints for the endomorphism action:

$$f: \mathrm{Hom}_\mathbb{Q}(F, \mathbb{C}) \to \mathbb{Z}_{\geq 0},\quad f(\sigma) = \dim_\mathbb{C}(\mathrm{Lie}(A_\mathbb{C})_\sigma)$$

Under the “simple signature” condition—i.e., for a distinguished CM type $\Phi$ and embedding $\sigma_1$ with $f(\sigma_1)=1$ and $f(\sigma)=0$ for all other $\sigma\in\Phi$—the ordinary reduction properties and Newton polygons of abelian varieties are sharply controlled by signature data (Farfán et al., 15 Aug 2025). The density of primes with ordinary reduction is $1$ (possibly after extension), and explicit densities are computed for higher-dimensional varieties with CM endomorphism fields. Examples include Jacobians of cyclic covers of projective lines, where signature varieties correspond to families with generic endomorphism and specific signature profiles.

8. Applications and Interdisciplinary Impact

The theory of signature varieties permeates several mathematical domains:

Context	Invariant/Variety	Role/Significance
Universal algebra/operads	Variety of signature Ω	Classifies algebraic structure/generators
Rough path theory	Signature tensor varieties	Encodes geometric path information
Time series, discrete	Discrete signature varieties	Time-warp invariant descriptor
Commutative algebra	F-signature varieties	Singularities, regularity detection
Noncommutative geometry	Curvature signature varieties	Links degree, curvature, eigenstructure
Arithmetic geometry	CM signature varieties	Organizes endomorphism, density results

Signature varieties support algorithms and software for symbolic and numerical computation (e.g., Macaulay2 packages for signature tensors (Améndola et al., 2 Jun 2025)). They offer tools for path learning, machine learning (via path-feature extraction), data analysis, singularity classification, and arithmetic distribution theorems.

9. Future Directions and Open Questions

Open problems include full characterization of discrete signature varieties, especially the analogues of the Chen–Chow theorem at higher tensor degrees and for arbitrary fields; further exploration of signature-based Hopf algebra structures in path geometry; extension of arithmetic density results to abelian varieties with non-simple signature or more general endomorphism algebras; and development of efficient symbolic/numeric frameworks for implicitization and invariant computation in high dimensions.

Signature varieties thus represent a unifying mathematical theme—at once a structural invariant, a geometric locus, and a computational object—whose ongoing paper defines and advances multiple contemporary research frontiers.