{"id":"83a829e8-ee50-4a6b-84fd-f3e441ed7b1b","arxiv_id":"2506.01429","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"PathSignatures is a new Macaulay2 package for computing signature tensors and signature varieties of piecewise polynomial paths, demonstrated on examples that match known results.","lead":"This paper presents a Macaulay2 software package, PathSignatures, that computes signature tensors and their algebraic varieties for piecewise polynomial paths. It includes tools for building paths, computing iterated-integral signatures, and exploring the associated algebraic varieties, with examples that reproduce published results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No mathematical error found in the worked examples; the central weakness is that PathSignatures is neither publicly released nor versioned, and the signature computation is only spot-checked, so the package claim cannot be independently verified.","rationale":"The strongest claim is a software capability claim. What would have to be true is that the package exists, runs, and produces correct exact algebraic objects. The paper's examples against AFS19 are genuine independent support: the universal variety U2,3 (dim 5, degree 4, six quadrics) and L3,3,2 (degree 90, 162 quadrics) are nontrivial and match published tables, and the numerical example for P2,4,3 and L2,4,3 also matches. I did not find a mathematical error in the derivations: the Lie-basis construction, shuffle algebra, Chen's formula, and the core-tensor parametrizations are standard and correctly set up on the displayed cases. The main soft spot is not the mathematics but the lack of a public, versioned release, which the reader already identified, coupled with the absence of any test suite or stated edge-case behavior for the signature computation. Since the package is the actual object being introduced, an unverifiable package is a legitimate reason for conditional rather than unconditional acceptance. I therefore keep the reader's CONDITIONAL verdict unchanged. Agreement is 'agree' because the reader's weakest assumption, namely exactness of the implementation beyond the tested examples, is exactly the point on which an independent randomized check should be run.","tokens_in":8295,"tokens_out":36969,"duration_ms":371149,"concrete_test":"Release a versioned PathSignatures package on a public repository and run the following in a clean Macaulay2 session: (1) re-run all examples in Sections 2-6 and compare outputs verbatim; (2) run a randomized test comparing sig() against an independent direct integration of the iterated integrals for, say, 100 random polynomial paths with degrees 1-8, dimensions 2-4, and 1-5 segments, including zero-length and repeated segments; (3) for 2-3 additional (d,k,m) triples not in the paper, compare the computed dimension and degree of P and L with NumericalImplicitization or AFS19 tables. Any mismatch would falsify the central claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that PathSignatures reliably computes signature tensors and signature varieties for piecewise polynomial paths. The evidence is a set of worked examples reproducing entries in AFS19 tables. I see no internal mathematical inconsistency in those examples: the Chen-Chow/universal-variety computation in Example 4.1, the core-tensor parametrizations in Section 6, and the polynomial-transform identity in Section 5 all check out on the displayed inputs. The load-bearing weakness is evidential rather than mathematical: the package is not released and no version identifier or repository is given, so the claim 'we introduce the Macaulay2 package' cannot be independently executed. In addition, the correctness of the signature engine is only demonstrated on a few curated cases; Section 2 states that 'the path signature is computed from the polynomial pieces through Chen's Formula' but does not specify the treatment of degenerate segments, zero-length pieces, repeated concatenations, or high-degree polynomials. A hidden implementation bug in those regimes would not be caught by the printed examples, yet would break the universal-quantifier claim about piecewise polynomial paths. This is why CONDITIONAL acceptance is appropriate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a Macaulay2 package, PathSignatures, for computing signatures of piecewise polynomial paths and the algebraic varieties associated with signature tensors. It defines a Path type for parametrized piecewise polynomial paths, computes signature tensors via Chen's formula, implements shuffle and half-shuffle products, Lyndon-word bases, and methods for universal varieties and core signature tensors. The package is demonstrated on several examples, including the implicitization of the universal variety U_{2,3}, the piecewise linear signature variety L_{3,3,2}, and numerical computations for P_{2,4,3} and L_{2,4,3}, with results matching tables from AFS19. The paper also states an equivariance relation for polynomial transformations of paths and verifies an instance of it.","tokens_in":726,"tokens_out":2627,"duration_ms":80085,"significance":"If the package is made publicly available and its signature engine is reliable, this is a useful software contribution to the algebraic statistics and nonlinear algebra community: it is the first dedicated Macaulay2 package for path signature varieties, and it combines exact symbolic implicitization with numerical fallback through NumericalImplicitization. The worked examples reproduce published dimensions, degrees, and ideals from AFS19, providing a credible sanity check for the computations. The mathematical content is largely computational rather than new theoretical results, but the package has the potential to lower the barrier for studying signature varieties. The examples are machine-checked transcripts, which is a strength: they are specific and reproducible in principle.","major_comments":[{"comment":"The central claim of the paper is the introduction of a Macaulay2 package, yet no code, repository, version identifier, or installation instructions are provided. The Macaulay2 session transcripts in the examples cannot be run or checked by the reader. For a software paper, the artifact is load-bearing: without it, the contribution is not independently verifiable. Please release the package with a persistent identifier and a version, and document its dependencies (NCAlgebra and NumericalImplicitization).","section":"Section 1"},{"comment":"The paper states that \"the path signature is computed from the polynomial pieces through Chen's Formula\" but gives no specification of the implementation's behavior for edge cases such as zero-length or degenerate segments, repeated concatenations, high-degree polynomials, or coefficient rings that are not fields. The examples in Sections 4 and 6 test only a few low-dimensional cases. Since the abstract and introduction claim support for all piecewise polynomial paths, this is a gap in evidence: a hidden implementation bug in one of these regimes would not be caught by the printed examples. Please provide a thorough test suite or a correctness argument for the signature engine, and specify the exact assumptions on the allowed inputs.","section":"Section 2"}],"minor_comments":[{"comment":"In Example 4.1, the line defining the ring Q is hard to parse: `Q = QQ new Array from (apply(lwords,i->y_i) | {Degrees => apply(lwords, i->length(i))});` uses `i` for a Lyndon word, which is confusing. Please use a different index name, such as `w`.","section":"Section 4"},{"comment":"In the sentence 'each coordinate σ_{i1,...,ik} of the tensor σ(k)(X) is a homogeneous polynomial of degree k', the notation σ_{i1,...,ik} is undefined. Consider defining it as the coefficient of the word i1...ik in the signature.","section":"Section 6"},{"comment":"The output of lyndonShuffle in Example 3.3 is a nested HashTable, which is hard to read. A more human-readable display would help, but this is a presentation issue.","section":"Section 3"},{"comment":"The authors should ensure that all code blocks are syntactically valid Macaulay2 code as printed; for instance, in Example 2.1 the line `i4 : M = id_(QQ^3);` is missing a space between `id` and `_` in the Macaulay2 convention (the printed form is acceptable, but please verify consistency).","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a software paper, and the absence of the actual code is a severe deficiency that prevents verification. The reader's report suggests conditional acceptance, but I believe the missing package and the lack of a correctness statement for the signature engine warrant major revision. The journal should require the authors to deposit the package in a public repository and include a test suite. The paper's mathematical claims are based on published results, and the examples match known values, so the underlying theory appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a clean, useful software paper. The genuinely new artifact is the Macaulay2 package PathSignatures, which is the first dedicated tool I know of for computing signature tensors and their associated algebraic varieties for piecewise polynomial paths. The design is sensible: the package handles parametrized families of paths, computes signatures via Chen's formula, implements Lyndon-word bases, shuffles, half-shuffles, and the Lie-algebra parametrization of the universal variety. The examples that reproduce dimension and degree data from AFS19 and CP20 are exact and internally consistent.\n\nWhat the paper does well: the worked examples are reproducible in principle, the connection between the code and the underlying theory is clear, and the numerical section (Example 6.2) sensibly integrates with NumericalImplicitization. The paper is honest about what it is: a software contribution with illustrative applications, not a new mathematical theorem. That is fine; the package fills a genuine gap.\n\nThe soft spots are proportionate. The main one is that the package is not publicly released and no repository or version identifier is given. The claim 'we introduce the Macaulay2 package' cannot be independently executed by a reader. This is not a mathematical flaw, but it is a real reproducibility issue. Second, the signature engine is verified only on a handful of curated cases. Section 2 says the signature is computed through Chen's formula but does not discuss degenerate segments, zero-length pieces, repeated concatenations, or high-degree polynomials. A hidden implementation bug in those regimes would not be caught by the printed examples. I do not think this is a fatal flaw; software papers often spot-check, and the displayed outputs match independent published tables.\n\nThe citation pattern looks fine. [AFS19] is a separate peer-reviewed paper and sharing an author is not a problem here because the reproduced tables are concrete benchmarks, not circular verifications. The references are appropriate and not padded.\n\nWho this is for: anyone working in algebraic statistics or nonlinear algebra on path signatures who wants to compute concrete examples. The paper deserves a serious referee with Macaulay2 expertise.\n\nMy recommendation: conditional acceptance. Ask the authors to make the package publicly available with a version identifier and to add a note on the treatment of degenerate segments. The mathematics in the examples holds up; the main fix is openness.","headline":"A solid software paper for Macaulay2 that fills a real niche; the math checks out on the worked examples, but the missing public release of the package is the main blocker.","tokens_in":9022,"tokens_out":583,"would_cite":true,"duration_ms":7555,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14Q15","15A69","60L10","60L70","62R01"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Macaulay2 package computes path signature varieties as ideals, dimensions, and degrees.","keywords":["path signatures","signature tensors","algebraic varieties","piecewise polynomial paths","Lyndon words","shuffle algebra","Macaulay2","algebraic statistics"],"falsifier":"Compute the signature of a piecewise polynomial path with a zero-length segment or a repeated concatenation and compare against direct numerical integration of the iterated integrals; a disagreement would show the Chen-formula implementation misses an edge case. Alternatively, use the package to compute the dimension and degree of a signature variety not in the literature tables, such as $L_{4,3,3}$ or $P_{3,5,4}$, and check against a numerical irreducible decomposition.","tokens_in":8106,"feed_emoji":"📐","tokens_out":7438,"duration_ms":71590,"temperature":0.7,"pith_summary":"PathSignatures is a Macaulay2 package for computing the signature tensors of piecewise polynomial paths and the algebraic varieties generated by those tensors. The signature of a path is a non-commutative power series of iterated integrals that almost determines the path; taking its fixed-degree parts produces signature tensors whose algebraic structure can be studied as varieties. The package represents a path by its polynomial segments, computes signatures through Chen's formula, manipulates words via the shuffle and half-shuffle products, builds the Lie algebra basis from Lyndon words, and returns the ring maps whose kernels define signature varieties. The authors verify the pipeline by reproducing the dimensions, degrees, and generator counts of published varieties such as $U_{2,3}$ and $L_{3,3,2}$, and they connect the package to numerical implicitization for cases where symbolic computation is too costly.","feed_headline":"Macaulay2 package computes path signature varieties","feed_subtitle":"Piecewise polynomial paths yield signature tensors and their varieties symbolically or numerically.","key_machinery":"The machinery centers on the signature map $\\sigma$ and three algebraic structures built around it. Chen's formula computes the signature of a concatenated path from the signatures of its pieces, which is what makes piecewise polynomial paths tractable. The shuffle product makes $\\sigma$ an algebra homomorphism to the coefficient ring, and the shuffle algebra is free commutative over Lyndon words; bracketings of those Lyndon words provide a basis of the free Lie algebra, so elements of $\\mathrm{Lie}_k(\\mathbb{R}^d)$ can be exponentiated by `tensorExp` to parametrize the universal variety $U_{d,k}$. For families of paths, the package precomputes the core tensors of the canonical axis and monomial paths and transports them by the diagonal action of a linear map.","core_discovery":"The paper claims that signature varieties of piecewise polynomial paths can be computed routinely in Macaulay2 by combining a few algebraic ingredients: Chen's formula for concatenating segment signatures, the shuffle product on the free associative algebra, Lyndon words as free commutative generators whose bracketings span the free Lie algebra, and the exponential map $\\exp$ from Lie coordinates to signature tensors. With these ingredients, the package produces parametrizations of the universal variety $U_{d,k}$ and of the polynomial and piecewise linear signature varieties $P_{d,k,m}$ and $L_{d,k,m}$, and implicitizes them to obtain defining ideals, dimensions, and degrees. The authors demonstrate this on examples, including $U_{2,3}$ (dimension 5, degree 4, six quadrics), $L_{3,3,2}$ (projective dimension 5, degree 90, one linear and 162 quadratic generators), and $P_{2,4,3}$ and $L_{2,4,3}$ (projective dimension 5, degrees 192 and 64), matching tables in [AFS19]. The package also implements the adjoint homomorphism for polynomial transformations of paths, so the signature of a transformed path can be obtained by evaluating an algebra map instead of computing new iterated integrals.","pith_inferences":["If the implementation is exact beyond the tested cases, the package is a ready data generator for conjectures about signature varieties, such as degree formulas or generator structures for larger $d$ and $k$.","The same Lyndon-word and shuffle-algebra pipeline could be ported to other computer algebra systems or to numerical libraries, because the underlying identities are independent of Macaulay2.","A natural next test is to exercise degenerate cases; the current examples do not include zero-length segments, repeated concatenations, or high-degree polynomials, so those behaviors remain unverified."],"forward_implications":["For any piecewise linear or polynomial family of paths, the package yields a parametrization of the associated signature variety, and from it the defining ideal, dimension, degree, and minimal generators.","When symbolic implicitization is infeasible, the package’s parametrizations plug directly into numerical implicitization routines, extending computations to larger $d$, $k$, and $m$.","Signatures under polynomial transformations of paths reduce to evaluating the adjoint word map, so no iterated integrals need to be computed anew for each transformed path.","The reproduced examples give users a checkable baseline: $U_{2,3}$, $L_{3,3,2}$, $P_{2,4,3}$, and $L_{2,4,3}$ all match published dimensions, degrees, and generator counts."],"supporting_citations":[{"why":"Gives Chen's formula used to compute the signature of a piecewise polynomial path from its polynomial segments.","marker":"[AFS19, Corollary 5.1]"},{"why":"Defines the universal variety $U_{d,k}$ as the projectivization of the level-$k$ projection of $\\exp(\\mathrm{Lie}_k(\\mathbb{R}^d))$, the object the package parametrizes.","marker":"[AFS19, Section 4]"},{"why":"Supplies the published dimension, degree, and quadrics for $U_{2,3}$ that the package reproduces.","marker":"[AFS19, Table 2]"},{"why":"Supplies the published dimensions and degrees for $L_{3,3,2}$, $P_{2,4,3}$, and $L_{2,4,3}$ that the package reproduces.","marker":"[AFS19, Table 3]"},{"why":"Provides the half-shuffle homomorphism and the adjoint relation for signatures of paths transformed by polynomial maps.","marker":"[CP20]"},{"why":"Establishes that the signature is an algebra homomorphism with respect to the shuffle product, which underlies the word-algebra computations.","marker":"[Ree58]"},{"why":"Provides the theorem that bracketings of Lyndon words form a basis of the free Lie algebra, used by `lieBasis`.","marker":"[Reu93]"},{"why":"States the Chen-Chow theorem that identifies the set of possible level-$k$ signature tensors with the projection of $\\exp(\\mathrm{Lie}_k(\\mathbb{R}^d))$, motivating the universal variety.","marker":"[FV10]"}],"fun_headline_variants":["Signature varieties of paths: now computable in Macaulay2","PathSignatures: new Macaulay2 package for signature varieties","New Macaulay2 package handles signature varieties of paths","Compute signature varieties of paths with Macaulay2 package"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim depends on the implementation of Chen's formula and the Lyndon-word shuffle algebra being exact for every piecewise polynomial path; the paper verifies this on selected examples but does not prove correctness or specify behavior for degenerate segments, high-degree polynomials, or repeated concatenations.","fun_headline_variants_meta":{"raw":{"variants":["Signature varieties of paths: now computable in Macaulay2","PathSignatures: new Macaulay2 package for signature varieties","New Macaulay2 package handles signature varieties of paths","Compute signature varieties of paths with Macaulay2 package"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001296,"raw_usage":{"total_tokens":5260,"prompt_tokens":887,"completion_tokens":4373,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":4302}},"tokens_in":503,"tokens_out":4373,"duration_ms":32243,"temperature":1.0,"reasoning_tokens":4302,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:42:28.450618+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the signature of a piecewise polynomial path with a zero-length segment or a repeated concatenation and compare against direct numerical integration of the iterated integrals; a disagreement would show the Chen-formula implementation misses an edge case. Alternatively, use the package to compute the dimension and degree of a signature variety not in the literature tables, such as $L_{4,3,3}$ or $P_{3,5,4}$, and check against a numerical irreducible decomposition.","supporting_citations":[],"review_version":1}