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Computing transcendence and linear relations of 1-periods

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read An algorithm now computes every rational linear relation among 1-periods, deciding transcendence and equality.

desk verdict First effective algorithm for 1-period relations, but the main theorem is conditional on a real gap in the supersaturation step. read the letter →

arxiv 2505.20397 v1 pith:4FN625OL submitted 2025-05-26 math.AG cs.SCmath.NT

Emre Can Sertöz , Joël Ouaknine , James Worrell This is my paper · ORCID
classification math.AGcs.SCmath.NT MSC 14Q0514C3014F4014H40
keywords 1-periodstranscendenceperiodrelationsmixedHodgestructuresJacobianmotivescomputationalalgebraicgeometryautonomousdifferentialequationsequalityproblem
topics The Hodge Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper presents an algorithm that, given finitely many 1-periods—integrals of univariate algebraic functions over algebraic data—computes a Q-basis for the space of Q-linear relations among them. From that basis it decides whether a single 1-period is transcendental and whether two 1-periods are equal, resolving the period-equality decision problem in the univariate case. The approach converts a theorem stating that all relations among 1-periods arise from 1-motives into explicit arithmetic on divisors on curves, using effective mixed Hodge structures and a saturation procedure. The same machinery yields an algorithmic classification of first-order autonomous differential equations.

What carries the argument

The load-bearing object is the Jacobian motive J^D_{C,E} of a punctured marked curve (C\D,E): a toric extension of the Jacobian of C marked by divisors supported on E, whose mixed Hodge structure is H1(C\D,E). The paper computes its Betti homology by rectilinear embedded graphs and its de Rham cohomology by second-kind differentials, then uses push-pull constructions to realise arbitrary 1-period motives as push-pulls of composites. The decisive identity is the theorem that for a saturated 1-motive, the space of period relations equals the expected relations generated by endomorphisms and trivial lattice and toric relations. The supersaturation procedure decomposes any Jacobian motive into a Baker motive—one with trivial abelian core, whose periods are combinations of 1, 2πi, and logarithms of algebraic numbers—and a saturated motive, transferring period relations by linear algebra.

What would settle it

Run the algorithm on the 1-periods log 2 = ∫$_1^{2}$ dx/x and log 4 = ∫$_1^{4}$ dx/x; correctness requires it to return exactly the one-dimensional relation space spanned by (2,−1), and a different answer, a false extra relation, or failure to terminate would refute the central claim.

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Extended reading notes

Core claim

The central claim is the Main Theorem: given representations (γ_i, ω_i) of 1-periods α_i, Algorithm 2.0.1 computes a Q-basis for the space rel_Q(α_1,...,α_k) of Q-linear relations. In particular, it decides whether a given 1-period is transcendental and whether two 1-periods are equal. The proof makes the dimension estimate for saturated 1-motives effective: it constructs the mixed Hodge structure H1(C\D,E) of punctured marked curves, builds the associated Jacobian motive, supersaturates it into a Baker motive plus a saturated motive, and reads off period relations from endomorphisms and trivial relations. This gives the first effective resolution of the period-equality problem for 1-periods.

Load-bearing premise

The algorithm is only guaranteed to stop if the internal routine that enlarges the motive until all endomorphisms are visible—a step relying on numerical approximation and lattice reduction—always terminates with exact output.

Editorial extensions

If this is right

  • Equality of any two 1-periods is decidable by a single uniform algorithm, rather than by case-by-case transcendence bounds.
  • Transcendence of a given 1-period can be certified: the algorithm returns a standard algebraic representation when the period is algebraic.
  • The Q-vector space of 1-periods is effective, meaning addition, scalar multiplication, and equality are all computable.
  • First-order autonomous differential equations P(u,u′)=0 over Q are algorithmically classified, refining the classical trichotomy by a toric and abelian type (a,b).
  • The effective mixed-Hodge-structure and correspondence subroutines provide reusable computational tools for studying algebraic curves and their periods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the main theorem is correct, transcendence tests for 1-periods no longer require explicit effective separation constants; the algorithm returns a finite exact description of all relations.
  • The refined-type routine for differentials suggests a natural test bed: implement the genus-zero and genus-one cases first, where known relations among logarithms and elliptic integrals provide immediate sanity checks.
  • The same effective mixed-Hodge-structure machinery could be pointed at higher-weight periods or at computing period lattices of families of curves, since the paper's subroutines are built for general punctured marked curves.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper presents Algorithm 2.0.1, which takes representations of 1-periods as integrals of rational differentials on algebraic curves and computes the space of Q-linear relations among them, thereby deciding transcendence and equality of 1-periods. The algorithm constructs the mixed Hodge structure of punctured marked curves, forms the associated Jacobian motive, supersaturates it, and then applies a description of period relations of saturated motives based on Huber–Wüstholz. The paper also gives an algorithmic classification of autonomous first-order differential equations. The main theorem is stated unconditionally but relies on several subroutines whose effectiveness is only partially justified.

Significance. If the main theorem holds, this is a major advance: it would be the first effective resolution of the Kontsevich–Zagier equality problem for 1-periods, including transcendence decisions. The paper contains a substantial amount of useful effective machinery: bases for algebraic de Rham cohomology, Betti homology via embedded graphs, push-pull Jacobian motives, explicit correspondences, and a reduction to divisor arithmetic on curves. It is also commendable that the paper cites deep external results (Huber–Wüstholz, Baker, Lombardo, CMSV) and is transparent about the places where arguments are only sketched. The additional classification of first-order autonomous differential equations is a nice application of the developed tools.

major comments (2)
  1. [§8.7, Lemma 8.7.54; §9.3.52] The splitting of the Baker component [K^χ_ψ → 0] from J^χ_{C,ψ} is not made effective. The proof of Lemma 8.7.54 explicitly says that making the split explicit by correspondences 'is possible but cumbersome' and instead proposes to put the period matrix into block form using effective Baker periods (8.8.60). No algorithm is given to recover an exact integral basis for H1([Z → 0]) from the numerical block form, nor is it shown that the resulting split is compatible with the lattice pullback ψ. This is load-bearing: the commutativity of Figure 1 in §9.3.52, used to compute the E-action on H1(J^χ_{C,ψ}), requires arrows 4 and 8 to be explicit maps on H1. Since these arrows are constructed using the non-effective splitting of Lemma 8.7.54, Step (2.0.9) of Algorithm 2.0.1 cannot be executed as written. The main theorem is therefore conditional on an effective construction of these splittings.
  2. [§8.3, Algorithm (8.3.32)] The termination of the Abel–Jacobi kernel computation is not rigorously established. The algorithm relies on LLL to 'guess' the kernel from numerical period approximations and on lower bounds for the rank of the Néron–Tate height matrix via certified nonzero minors. The text asserts that 'LLL will eventually find a lattice of full rank in KD' but gives no quantitative convergence argument (e.g., a lower bound on the gap between the period lattice and the kernel). Since KD is a direct input to the Baker splitting in (8.4.39) and (8.6.46), and hence to the supersaturation process, this is another load-bearing point that needs a proof or a precise reference.
minor comments (6)
  1. [§9.2.20] Typo: 'On could imagine' should be 'One could imagine'.
  2. [§1.1.17] Typo: 'determening' should be 'determining'.
  3. [§3.5.37] The equation 'h0(Ω1C((−d + 1)p) = 0' is missing a closing parenthesis; it should read 'h0(Ω1C((−d + 1)p)) = 0'.
  4. [§5.8.62] The asterisk in the block matrix (5.8.62.1) is not defined; a sentence explaining the entries would help.
  5. [§8.3.31] In (8.3.31.1), the notation '(F^1 H^1_AdR(C,D))^∨ = (F^1 H^1_AdR(C))^∨' seems to identify a quotient with a subspace; the intended identification should be stated.
  6. [§8.8.61] The reference to a 'day-and-night algorithm analogous to the one given in §8.3' would benefit from a precise pointer to the termination argument.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the central derivation relies on external theorems (Huber-Wüstholz, Baker, CMSV18/Lombardo), and the only self-citation is non-load-bearing.

full rationale

The paper's main algorithm computes the Q-linear relation space rel_Q(α_1,...,α_k) as the pullback I^{-1}R(M) of a period-relation space computed from motives (2.0.13). The chain is: represent H^1_{dR}(C\D,E) and H^B_1(C\D,E) from explicit curve and chain data (§§3–4); attach Jacobian and push-pull motives (§6); import End(J_C) unconditionally from Lombardo/CMSV18 (7.2.27); supersaturate via divisor arithmetic and correspondences (§9); compute Baker relations from Baker's theorem on logarithms (8.8.61); and identify saturated-motive relations with expected relations via Theorem 5.6.49, which is explicitly a reinterpretation of the external Huber–Wüstholz Dimension Estimate [HW22, Thm 15.3]. No step defines the target relation space in terms of itself, and no fitted parameter is renamed as a prediction. The single self-citation, [CS21] in Remark 7.2.24, concerns a practical upper bound that the paper itself marks as expected and that is not used in the unconditional termination proof; it is therefore not load-bearing. The internal gap flagged at Lemma 8.7.54 — where an explicit correspondence splitting is replaced by numerical block-form period reduction — is a completeness and termination risk, not circularity: it leaves the algorithm conditional, but it does not make the claimed output equivalent to the input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim does not introduce new entities or fitted constants. All inputs are external algorithms and theorems.

assumptions (4)
  • domain assumption Wüstholz's analytic subgroup theorem and the Huber-Wüstholz dimension estimate for saturated 1-motives [HW22, Theorem 15.3].
    The theorem that R(M)=Re(M) for saturated motives (Theorem 5.6.49) is the theoretical core of the algorithm; without it the period relation computation collapses.
  • domain assumption Effective algorithms for computing endomorphism algebras of Jacobians [CMSV18; Lom18].
    The algorithm requires End(JC) to compute endo-relations and to perform supersaturation. [CMSV18] is conditional on the Mumford-Tate conjecture for termination; [Lom18] provides an unconditional but impractical alternative, keeping the main theorem unconditional.
  • standard math Baker's theorem on linear forms in logarithms of algebraic numbers [Bak66].
    Used in Lemma 8.8.61 to compute the relations among Baker periods, reducing them to multiplicative relations among algebraic numbers.
  • domain assumption Certified arbitrary-precision numerical integration and comparison isomorphisms (e.g., [Mez16; LMS19]).
    The Betti-to-de Rham comparison is approximated numerically; exact integer homology coordinates are recovered from ball arithmetic, assuming the precision can be increased until the integer vector is unique.

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Cite this review

Pith. "Pith review of Computing transcendence and linear relations of 1-periods." pith.science (2026). https://pith.science/paper/4FN625OL

@misc{pith2026250520397,
  author       = {Pith},
  title        = {Pith review of: Computing transcendence and linear relations of 1-periods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4FN625OL}},
  note         = {Machine review of arXiv:2505.20397}
}
read the original abstract

A 1-period is a complex number given by the integral of a univariate algebraic function, where all data involved -- the integrand and the domain of integration -- are defined over algebraic numbers. We give an algorithm that, given a finite collection of 1-periods, computes the space of all linear relations among them with algebraic coefficients. In particular, the algorithm decides whether a given 1-period is transcendental, and whether two 1-periods are equal. This resolves, in the case of 1-periods, a problem posed by Kontsevich and Zagier, asking for an algorithm to decide equality of periods. The algorithm builds on the work of Huber and W\"ustholz, who showed that all linear relations among 1-periods arise from 1-motives; we make this perspective effective by reducing the problem to divisor arithmetic on curves and providing the theoretical foundations for a practical and fully explicit algorithm. To illustrate the broader applicability of our methods, we also give an algorithmic classification of autonomous first-order (non-linear) differential equations.

Figures

Figures reproduced from arXiv: 2505.20397 by the authors.

Figure 1
Figure 1. Making the action on a supersaturated motive effective. (9.3.44) The right hand side. We will describe the arrows appearing on the right hand side of [PITH_FULL_IMAGE:figures/full_fig_p062_1.png] view at source ↗

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Works this paper leans on

4 extracted references · 2 canonical work pages

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    work page arXiv 1954

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    On the differential equations satisfied by period matrices,

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    doi: 10.1007/s00222-005-0440-5. [GF] J. I. B. Gil and J. Fresán, “Multiple zeta values: From numbers to motives,” in Clay Mathematics Proceedings, to appear. [GH94] P. Griffiths and J. Harris,Principles of Algebraic Geometry. Wiley, 1994.doi: 10.1002/9781118032527. [Gro66] A. Grothendieck, “On the De Rham cohomology of algebraic varieties,”Publica- tions ...

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