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The continuous symmetries of a variety given by a rational parametrization can be read off the parametrization itself, without ever computing its equations.

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2026-07-12 07:46 UTC pith:37XAL336

load-bearing objection Clean geometric characterization of the symmetry Lie algebra from a parametrization alone, plus a usable Monte-Carlo algorithm that never needs the ideal.

arxiv 2607.02676 v1 pith:37XAL336 submitted 2026-07-02 math.AG cs.CCmath.RT

Computing the continuous symmetries of a parametrized variety

Benjamin Biaggi , Jan Draisma , Fulvio Gesmundo , Aida Maraj , Magdal\'ena Mi\v{s}inov\'a This is my paper
classification math.AG cs.CCmath.RT MSC 15A8617B4568W3014Q20
keywords symmetry Lie algebraunirational varietyparametrized varietyGL-binomial idealMonte Carlo algorithmstaged tree modelscolored Gaussian graphical modelssecant varieties
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

When an algebraic variety arrives only as the image of a rational map, most classical invariants require first finding the ideal of polynomials that vanish on it—an expensive step that is often intractable. This paper shows that the continuous linear symmetries of that image, encoded by its Lie algebra, can be recovered directly from the map and its Jacobian. The resulting Monte-Carlo algorithm runs in time polynomial in the bit-length of the input and succeeds with arbitrarily high probability. The same machinery decides whether a linear change of coordinates can make the ideal binomial, and is applied to staged-tree models, colored Gaussian graphical models, rational curves and secant varieties. The practical consequence is that a large class of symmetry and binomiality questions that previously demanded Gröbner bases become feasible from a mere parametrization.

Core claim

The symmetry Lie algebra g_X of a variety X that is the image of a rational map φ equals the set of all linear maps A such that A·φ(p) lies in the column space of the Jacobian of φ at every sufficiently general parameter point p. Consequently a finite sample of such points yields a linear system whose solution space is exactly g_X, and this system can be solved by a polynomial-time Monte-Carlo procedure.

What carries the argument

The geometric characterization (Theorem 2.2 / Corollary 2.3) that A belongs to g_X if and only if A sends every point of a dense open set of X into the tangent space at that point, which for a parametrized variety is the column space of the Jacobian; this identity is the sole engine of the Monte-Carlo algorithm and of all subsequent applications.

Load-bearing premise

The proof that a polynomial-size sample of random points already cuts out the full symmetry algebra relies on a crude upper bound on the degree of a certain hypersurface; if that degree is much larger than claimed, the success probability guarantee fails.

What would settle it

Run Algorithm 1 on a family of parametrizations whose symmetry Lie algebras are known by independent means (e.g., Veronese curves or matrix pencils) and check whether the returned basis matches the known algebra for sample sizes predicted by the degree bound; systematic under-estimation of the algebra would falsify the analysis.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proves that the symmetry Lie algebra g_X of a unirational variety X = im(φ) can be recovered from the parametrization alone: g_X consists of those A in gl_n such that A·φ(p) lies in the column space of the Jacobian J_φ(p) for all p in a Zariski-open set U_φ (Theorem 2.2, Corollary 2.3). From this characterization the authors derive a polynomial-time Monte-Carlo algorithm (Algorithm 1 / Theorem 1.1) that returns a basis of g_X ∩ gl_n(Q) with success probability ≥ 1−ε, controlled by an explicit Schwartz–Zippel bound, together with a companion Monte-Carlo test for GL-binomiality (Algorithm 2 / Corollary 1.2). Deterministic variants, applications to staged-tree models and colored Gaussian graphical models, a complete classification of nonzero symmetry algebras of rational curves (Theorem 7.2), and a comparison of symmetry algebras of a variety and its secants under an expected-dimension hypothesis (Theorem 8.1) complete the work. Code and experimental data are released on GitHub.

Significance. The geometric characterization and the resulting Monte-Carlo algorithms remove the need to compute a vanishing ideal before determining continuous linear symmetries—an operation that is often the bottleneck for unirational models arising in algebraic statistics. The algorithms are fully specified, run in polynomial time under standard encodings, and are accompanied by open-source implementations that already produce new examples (non-GL-binomial staged trees of depth 3, counter-examples to a conjecture on BMT-derived models). The curve classification and the secant comparison are clean theoretical by-products that will be useful beyond the algorithmic setting. The contribution is therefore both practical and theoretically solid for the intended audience.

minor comments (5)
  1. In the proof of Proposition 3.1 the degree bound 2n³d(2m+1) is obtained by deliberately crude estimates on minors. While sufficient for the polynomial-time claim, a short remark that the bound is not sharp (and that practical implementations may safely use smaller N) would help readers who implement the algorithm.
  2. Algorithm 2 returns a Boolean answer; Remark 4.4 sketches how to recover an explicit change of coordinates when the answer is true. A pointer to a concrete simultaneous-diagonalization routine (or a note that the splitting-field degree may be exponential) would make the remark more self-contained.
  3. Tables 1–4 list dimensions of Lie algebras after linear relations are removed, but the precise reduction step is only alluded to in Remark 3.5. A one-sentence clarification in the table captions would improve readability.
  4. The deterministic Algorithms 3 and 4 are correct but lose polynomiality because of expression swell. A brief warning that intermediate rational functions can become super-polynomial in size would set expectations for users who try them on larger examples.
  5. A few typographical inconsistencies appear (e.g., “T esting” in the section heading, occasional missing spaces around math operators). A light copy-edit pass would remove them.

Circularity Check

1 steps flagged

No load-bearing circularity; main Lie-algebra characterization and Monte-Carlo algorithm rest on classical algebraic-group facts plus an independent geometric argument, with only incidental self-citations for context.

specific steps
  1. self citation load bearing [Section 2, Proposition 2.1 and proof of Theorem 2.2]
    "We recall the following result which is a combination of [GHL25, Lemma 3.1] and [Bor91, Lemma 7.4] … g_X = {A ∈ gl_n | A.f ∈ I(X) for all f ∈ I(X)}. … Using this characterization at the level of ideals, we obtain the following infinitesimal characterization …"

    The ideal-level description of g_X is taken from a paper sharing an author (GHL25). While the subsequent tangent-space reformulation is new and elementary, the load-bearing starting point is a self-citation rather than a fully external classical reference. The circularity is minor: Borel supplies an independent classical foundation, and the rest of the paper does not rely on any uniqueness or ansatz from GHL25.

full rationale

The core claim (Theorem 2.2 / Corollary 2.3) equates g_X with the set of linear maps sending points of a dense set into the corresponding tangent spaces (or Jacobian column spaces). Its short proof invokes only the standard ideal-level characterization of the Lie algebra of an algebraic group (Prop. 2.1, combining Borel with a lemma from GHL25) and the elementary observation that A.f vanishes on X precisely when A.q lies in T_q X. No free parameters are fitted, no uniqueness theorem is imported from the authors’ prior work to force the result, and the subsequent Monte-Carlo analysis (Prop. 3.1) is a self-contained Schwartz–Zippel degree bound. Self-citations (GHL25, MP26, KV25, BDM26) supply background, experimental motivation or dual statements, but none is required for the correctness of the geometric characterization or the polynomial-time claim. Applications to staged trees, colored Gaussians, rational curves and secants are independent verifications or corollaries, not circular reductions. Hence the paper is essentially free of the six circularity patterns; the single minor self-citation raises the score only to 1.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The work is pure mathematics over C (or Q). It relies only on standard facts from algebraic geometry and algebraic groups; no free parameters are fitted and no new physical or statistical entities are postulated. The sole non-standard ingredients are the explicit degree bounds used for the probabilistic analysis and the technical non-degeneracy assumptions on the input encoding.

axioms (4)
  • standard math Zariski topology coincides with Euclidean topology for the closures under consideration; tangent spaces and Lie algebras of algebraic groups behave as in Borel’s Linear Algebraic Groups.
    Invoked throughout Sections 2–4 and in the proofs of Theorems 2.2 and 8.1.
  • standard math Schwartz–Zippel lemma applies to the hypersurface of degree ≤ 2 n³ d (2m+1) that contains the non-generic sample points.
    Used in the correctness proof of Algorithm 1 (Proposition 3.1).
  • domain assumption Input rational functions are given either in standard sparse encoding or as algebraic circuits over Q, with every variable appearing and at least one non-zero function.
    Technical hypotheses stated before Proposition 3.2 that guarantee polynomial running time.
  • domain assumption A maximal torus of the identity component of the symmetry group has a dense orbit on X if and only if X is GL-binomial.
    Taken from KV25 / classical toric geometry and used as the decision criterion in Algorithm 2.

pith-pipeline@v1.1.0-grok45 · 32018 in / 2493 out tokens · 20193 ms · 2026-07-12T07:46:57.574350+00:00 · methodology

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read the original abstract

We prove that the symmetry Lie algebra of a parametrized variety can be determined directly from the parametrization, without computing the vanishing ideal of the variety. We derive a practical polynomial-time Monte Carlo algorithm for computing the symmetry Lie algebra of a parametrized variety. We discuss applications to testing the binomiality of the ideal of a parametrized variety after changing coordinates, and test this property on varieties arising from staged tree models and colored Gaussian graphical models. Finally, we discuss symmetries and binomiality after changing coordinates for rational curves and give a characterization of the symmetries of many secant varieties.

Figures

Figures reproduced from arXiv: 2607.02676 by Aida Maraj, Benjamin Biaggi, Fulvio Gesmundo, Jan Draisma, Magdal\'ena Mi\v{s}inov\'a.

Figure 1
Figure 1. Figure 1: A stage tree model realized by the rational normal curve of degree n − 1. The matrices D′ =         0 −1 −1 · · · −1 −1 0 1 −1 · · · −1 −1 0 0 2 · · · −1 −1 0 0 0 . . . . . . . . . 0 0 0 · · · n − 2 −1 0 0 0 · · · 0 n − 1         and B ′ =         −1 −1 −1 · · · −1 −1 1 −1 −1 · · · −1 −1 0 2 −1 · · · −1 −1 0 0 3 · · · −1 −1 . . . . . . . . . . . . . . . . . . 0 0 0 · · · n − 1 −1  … view at source ↗

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