Pith. sign in

REVIEW 3 major objections 2 minor 15 references

Special Functions and Geometries in Scattering Amplitudes: From Particle Physics to Gravity

T0 review · 3 major / 2 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A Calabi-Yau threefold appears for the first time in gravitational-wave integrals at fifth post-Minkowskian order, and that integral is solved analytically; the same thesis obtains the first symbol of the twelve-point elliptic double box.

desk verdict A solid, clearly-written compilation thesis; the underlying results are real, but the 'first CY threefold at 5PM' claim rests on an asserted completeness of the Baikov topology enumeration. read the letter →

arxiv 2506.11911 v1 pith:I2LS72VM submitted 2025-06-13 hep-th gr-qc

Roger Morales This is my paper
classification hep-thgr-qc PACS 04.30.-w
keywords FeynmanintegralsscatteringamplitudesellipticcurvessymbolbootstrapSchubertanalysispost-MinkowskianexpansiongravitationalwavesCalabi-Yauthreefold
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

This thesis tries to establish which geometries and special functions actually occur in Feynman integrals, in two very different settings. In planar N = 4 super Yang-Mills theory it claims that two ten-point ladder families share one and the same elliptic curve at every loop order, and, by generalizing the Schubert analysis to elliptic integrals, it produces the first symbol of the two-loop twelve-point elliptic double box as a compact one-line formula. In post-Minkowskian gravity it claims a systematic census, based on the Baikov representation and leading singularities, of the geometries appearing in two-body black-hole scattering up to four loops, identifying the first Calabi-Yau threefold relevant to gravitational-wave physics at fifth post-Minkowskian order. It then solves the integral behind that Calabi-Yau geometry by bringing its differential equation into epsilon-factorized (canonical) form, which required a new method that accommodates epsilon-dependent apparent singularities. If these claims are right, analytic calculations at the current precision frontier, gravitational-wave templates and multi-loop SYM amplitudes, can proceed through geometries beyond polylogarithms with a known function language.

What carries the argument

The elliptic half runs on the elliptic symbol: the symbol of an elliptic multiple polylogarithm is a tensor product of ordinary logarithms and torus images (periods of the elliptic curve), which turns the bootstrap into linear algebra on letters. The Schubert analysis, solving the momentum-twistor problem of which line in $\mathbb{CP}^3$ intersects four given external lines, generates those letters, including new elliptic last entries, and the integrability conditions plus the hexagon differential equation fix all coefficients. The gravity half runs on the loop-by-loop Baikov representation, which rewrites an integral in its propagator variables so that the maximal leading singularity exposes the underlying algebraic curve, surface, or Calabi-Yau variety, cross-checked by the factorization of the Picard-Fuchs operator. To solve the Calabi-Yau integral, the load-bearing mechanism is the canonical (epsilon-factorized) form of the differential equation, $\vec J' = \varepsilon A(x)\vec J$, whose construction requires a new transformation that removes apparent singularities depending on the dimensional regulator $\varepsilon$.

What would settle it

An independent, exhaustive enumeration of the four-loop (fifth-post-Minkowskian) integral topologies that finds one missing diagram whose leading singularity is a geometry absent from the thesis's classification, a genus-two curve or a Calabi-Yau of a different dimension, say, would falsify the 'first Calabi-Yau threefold' claim; for the solution half, a high-precision numerical evaluation of the 2SF Calabi-Yau integral at fixed kinematics that disagrees with the $\varepsilon$-factorized formula would falsify it.

Watch

Extended reading notes

Core claim

The central claim is that the geometries of Feynman integrals can be predicted before any calculation and then handled analytically. On the SYM side, the same elliptic curve governs two 10-point ladder families to all loop orders, and the symbol of the 12-point elliptic double box, previously out of reach because direct integration fails, is fixed uniquely by integrability conditions together with the differential equation to the one-loop hexagon, giving the one-line symbol formula of eq. (3.64). On the gravity side, the claim is that a loop-by-loop Baikov leading-singularity analysis classifies all geometries of the post-Minkowskian two-body scattering up to four loops, with the first Calabi-Yau threefold in gravitational-wave physics appearing at fifth post-Minkowskian order; the corresponding integral is then solved by finding an epsilon-factorized differential equation even though the naive equations carry epsilon-dependent apparent singularities, via a new transformation method developed for that purpose.

Load-bearing premise

The 'first Calabi-Yau threefold at fifth post-Minkowskian order' claim rests on the assumption that the Baikov leading-singularity census enumerated every contributing integral topology and subsector and classified their geometries correctly; that completeness is asserted rather than proved in Chapter 4.

Editorial extensions

If this is right

  • Post-Minkowskian calculations at and beyond fifth order know in advance which function classes they must confront: polylogarithms below 5PM, then elliptic and K3 geometries, and a Calabi-Yau threefold at 5PM, so no integral should arrive as an unexpected function class.
  • The epsilon-factorized form for the 2SF Calabi-Yau integral makes that integral evaluable order by order in $\varepsilon$ via path-ordered exponentials, which is the step needed for analytic 5PM gravitational-wave predictions in that sector.
  • The symbol of the twelve-point elliptic double box completes the symbol dictionary of the two-loop planar basis in N=4 SYM (together with the pentabox and double pentagon), so all two-loop planar SYM amplitudes are symbolically accessible.
  • The two 10-point elliptic ladder families provide all-loop-order laboratories where every rung returns to the same elliptic curve, suitable for stress-testing new elliptic integration and bootstrap tools as they are developed.
  • The new method for $\varepsilon$-dependent apparent singularities extends the reach of canonical-form techniques beyond the polylogarithmic, elliptic, and previously known Calabi-Yau cases, since those methods could not cope with such singularities.

Reading between the lines

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

  • If the classification is complete, the same Baikov census could be rerun for other gravitational observables, such as spin-dependent terms, eccentric orbits, or radiative waveform integrals, and there is no guarantee the first non-polylogarithmic geometry appears at the same loop order there.
  • The elliptic symbol bootstrap fixed its entire ansatz from integrability alone plus one differential equation; that suggests the same strategy could crack other multi-scale elliptic integrals whose function class is unknown, including massive or non-planar cases.
  • The apparent-singularity method is likely generic: $\varepsilon$-dependent apparent singularities are expected to be common in higher-loop Calabi-Yau integrals, so the technique probably transfers to banana integrals and QCD-style multi-scale problems beyond the specific 2SF integral.
  • One consequence the author does not push: the compact form of the 12-pt double-box symbol reads like an integrated version of the hexagon differential equation, hinting that symbol-level integration may be the natural language for elliptic letters, not just polylogarithmic ones.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. This PhD thesis collects published work in two areas. In the N=4 SYM part, it introduces two ten-point ladder families that are claimed to involve the same elliptic curve to all loop orders, provides one-fold elliptic integral representations for them, and initiates an elliptic symbol bootstrap that produces a compact symbol formula for the two-loop twelve-point elliptic double box. In the gravitational-wave part, it classifies Feynman-integral geometries in the post-Minkowskian expansion through four loops using loop-by-loop Baikov leading singularities, identifies a Calabi-Yau threefold at fifth post-Minkowskian order, and derives an epsilon-factorized differential equation for the corresponding integral despite the presence of epsilon-dependent apparent singularities. The thesis is based on the author's published papers [1]-[6] and includes extensive consistency checks of the main computations.

Significance. If the 'first Calabi-Yau threefold at 5PM' claim survives scrutiny, it is a significant result for gravitational-wave amplitude technology, since it delimits the class of special functions needed at the current PM frontier. The two-loop twelve-point double-box symbol is a first example of an elliptic symbol bootstrap and provides a concrete target for elliptic symbol-level integration, with an independent check already cited in ref. [218]. The two ten-point ladder families are useful testbeds for elliptic polylogarithm technology and their explicit three-loop evaluation is a concrete computational achievement. The thesis is strong on verification: the 12-pt symbol is checked against integrability, Steinmann conditions, the hexagon differential equation, symmetry limits, a conformal Ward identity, and the soft limit to the 10-pt double box; the PM classification is cross-checked against known results and, in part, against Picard-Fuchs operators.

major comments (3)
  1. [Sec. 4.3.1 and Sec. 4.8, with Sec. 2.6 as a counterexample] The headline claim that a Calabi-Yau threefold appears for the first time at fifth post-Minkowskian order rests on the completeness of the Baikov leading-singularity classification for every contributing topology and every subsector. That completeness is asserted rather than proved. The thesis itself provides a warning example in Sec. 2.6: the staggered elliptic ladder family has an algebraic leading singularity in the top sector, with the elliptic curve inherited from a subsector, and the text explicitly leaves the all-subsector check to future work. Because the same top-sector maximal-cut diagnostic is the main tool of Chapter 4, a missed topology, a mishandled ISP residue, or a nontrivial subsector geometry could move the 'first' appearance to a different PM order. Please either supply a complete all-subsector and ISP analysis for the PM classification, or state the headline claim as conditional on the completeness of the enumeration with the class of checked sectors precisely delimited.
  2. [Abstract and Secs. 2.5-2.8] The advertised result that the two ladder families 'involve the same elliptic curve to all loop orders' is stronger than the evidence presented in the thesis. Section 2.8 states that elliptic linear reducibility and the leading-singularity invariance were checked only up to six loops, and that a general proof is expected in the future. The one-fold representations in eqs. (2.94) and (2.95) are therefore established at finite loop order, not at all loop orders. Please either weaken the abstract and introductory claims to 'verified up to six loops with strong evidence for all loops' or provide an induction argument that upgrades the pattern to a proof. As written, the central claim of Part II overstates what is demonstrated.
  3. [Sec. 3.5.1] The bootstrap fixes more than 400,000 coefficients by solving integrability conditions numerically at random kinematic points using high-precision row reduction. The text reports the number of points and the run time but does not state that the resulting coefficients were subsequently verified to satisfy the integrability polynomials symbolically over the full kinematic domain. If the verification is only numerical, accidental vanishing at the sampled points cannot be excluded in principle. Please specify the exact procedure that turns this numerical computation into a rigorous statement, for example exact rational reconstruction, algebraic independence arguments, interpolation in the cross-ratios, or reliance on the independent check of ref. [218].
minor comments (2)
  1. [Eq. (3.64)] The notation '-(chi14 -> infinity)' in eq. (3.64) is informal: the first term is formally divergent in that limit, and the intended subtraction is only understandable through the preceding limiting discussion in eqs. (3.59)-(3.63). Please define the subtracted term explicitly as a limit.
  2. [Computational reproducibility] The thesis would benefit from a reproducibility note: for the twelve-point bootstrap and the three-loop ladder evaluation, links to the actual Mathematica notebooks or ancillary files should be provided so that the numerical row reductions, the coefficient fixing, and the reported run times can be audited independently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is constrained by integrability conditions, external differential equations, and independent cross-checks, not by fitted inputs or self-citation loops.

full rationale

Walking the paper's derivation chain, no step reduces by construction to its own input. In the SYM half, the 12-point elliptic double-box symbol is not fitted to known data: the Schubert analysis supplies a candidate symbol alphabet, and the ansatz is then fixed by 100 integrability constraints plus the external differential equation relating the double box to the hexagon (Eq. 3.8), leaving only an overall scale that is fixed by that same differential equation; the result is subsequently checked against soft limits, Steinmann conditions, symmetries, and the conformal Ward identity. In the gravity half, the classification of PM geometries uses loop-by-loop Baikov leading singularities (Sec. 4.3.1) with IBPs and Picard-Fuchs operators as cross-checks rather than as fitted parameters, and the 'first Calabi-Yau threefold at 5PM' claim is presented as a consequence of that classification. The thesis is based on the author's own published papers, and self-citation is frequent, but the load-bearing arguments are restated and cross-checked within the thesis itself and against external results; none of the central claims is justified solely by an unverified self-citation. The skeptic's concern—that the 5PM 'first' claim assumes completeness of the enumerated topologies and correct handling of all subsector/ISP contributions—is a genuine correctness risk about an unproven completeness assumption, not a circularity: no equation is defined in terms of the claimed conclusion, and no fitted quantity is renamed as a prediction. Accordingly, no circular step can be exhibited, and the appropriate score is 0.

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

No free parameters are fitted to data; kinematic cross-ratios are external variables. No new physical entities are introduced. The Calabi-Yau threefold is an established mathematical object newly found in a physical setting; the elliptic symbol letters and apparent singularity method are new tools, not entities.

assumptions (4)
  • domain assumption The leading singularity (maximal cut) in the Baikov representation captures the underlying geometry of a Feynman integral.
    Used throughout Ch. 4 to classify PM integral geometries; the thesis notes subtle cases involving ISP residues and rationalization, but relies on this for the classification.
  • domain assumption The set of Feynman integral topologies considered in the PM classification up to five post-Minkowskian order is complete.
    Needed for the conclusion that the identified Calabi-Yau threefold is the first in gravitational-wave physics; no proof of completeness is given.
  • ad hoc to paper The Schubert analysis generates the complete symbol alphabet for the 12-pt double box.
    The bootstrap in Ch. 3 fixes coefficients within this alphabet; if letters are missing, the unique solution from integrability would not be the true symbol.
  • domain assumption The elliptic curve underlying the 10-pt and 12-pt double boxes is correctly identified by the 2-loop Schubert problem.
    Needed for the last entries of the elliptic symbol; cross-checked by the differential equation with the hexagon.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Special Functions and Geometries in Scattering Amplitudes: From Particle Physics to Gravity." pith.science (2026). https://pith.science/paper/I2LS72VM

@misc{pith2026250611911,
  author       = {Pith},
  title        = {Pith review of: Special Functions and Geometries in Scattering Amplitudes: From Particle Physics to Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2LS72VM}},
  note         = {Machine review of arXiv:2506.11911}
}
read the original abstract

This thesis focuses on the fields of scattering amplitudes and Feynman integrals, with an emphasis on the geometries and special functions that they involve, and is devoted to two distinct research directions. In the first half of the thesis, we explore elliptic Feynman integrals in maximally supersymmetric Yang-Mills (N = 4 SYM) theory. In particular, we study elliptic generalizations of ladder diagrams, identifying the first two families of Feynman integrals involving the same elliptic curve to all loop orders, which provide a great testing ground for developing mathematical tools that can facilitate the calculation of elliptic integrals. In this direction, we initiate the symbol bootstrap for elliptic Feynman integrals, for which we generalize the so-called Schubert analysis to predict elliptic symbol letters. As a proof of principle, we obtain for the first time the symbol of the two-loop twelve-point elliptic double-box integral, resulting in a compact, one-line formula. In the second half of the thesis, we pioneer a systematic investigation of the Feynman integral geometries that are relevant to the study of gravitational waves emitted during the inspiral phase of black-hole mergers within the post-Minkowskian expansion of classical gravity. Specifically, we classify the geometries and special functions appearing in the expansion up to four loops. Among other findings, we identify the first Calabi-Yau three-dimensional geometry relevant to gravitational-wave physics. Subsequently, we study the Feynman integral giving rise to this Calabi-Yau geometry in more detail, and solve it by bringing its differential equation into canonical form - a step that required developing a new method that accommodates for apparent singularities.

Figures

Figures reproduced from arXiv: 2506.11911 by the authors.

Figure 1.1
Figure 1.1. Diagram for the two-loop four-point massless box-bubble, which is used in the main text to exemplify the concepts of integral family, sectors and subsectors. where the external momenta pi are all outgoing, and pi,...,j ≡ pi + ⋅ ⋅ ⋅ + pj . In general, Feynman integrals can also include a numerator factor N (ki ⋅ kj , ki ⋅ pj) that may depend on the L(L+1)/2 scalar products ki ⋅kj between loop momenta, and on the EL p… view at source ↗
Figure 1.2
Figure 1.2. Hyperbola P(z) = √ z 2 − 1 (shown in black) and line L(z) = t(z + 1) (shown in red), used in the main text to exemplify the principle of rationalization of square roots. integrals in D = 4 [48,56] generalizing the double box; see tab. 1.1 for reference. Notice, however, that there are also integrals such as the L-loop tardigrade in D = 4, which involve a CY manifold of dimension 2L − 2 [49, 53], which grows much fas… view at source ↗
Figure 1.3
Figure 1.3. Examples of closed contours for eq. (1.59), in the case of (a) a quadratic polynomial, and (b) a quartic polynomial. without further transcendental integrals left, then the integral admits a d log form. This is related to the previous subsection, where we demanded that Ln fully factorized into L1 ⋯L1, since the solutions to the latter are given by algebraic functions. Otherwise, if the leading singularity contains n… view at source ↗
Figures from the paper (23 more)
Figure 2.1
Figure 2.1. Figure 2.1: Momentum-space diagram (in black) and dual graph (in blue) for the L-loop (4L + 4)-pt traintrack diagram, where dashed lines indicate an arbitrary number of loops. where the ± comes from the choice of sign for the square root in eq. (2.13). Alternatively, we could ha…
Figure 2.2
Figure 2.2. Figure 2.2: (a) Lattice Λ = Zω1 + Zω2, spanned by the periods ω1 and ω2. The area in gray defines the fundamental parallelogram. (b) Location of the roots ri of the elliptic curve in kinematic space, and integration contours γj used in the main text to define Abel’s map and the …
Figure 2.3
Figure 2.3. Figure 2.3: Momentum-space diagram (in black) and dual graph (in blue) for an (R +L)-loop diagram in the 10-pt elliptic ladder family, which has R (L) loops to the right (left) of the rung with external legs p5 and p10. The dashed lines indicate an arbitrary number of loops. Due…
Figure 2.4
Figure 2.4. Figure 2.4: Momentum-space diagram (in black) and dual graph (in blue) for an (R + M + L)-loop diagram in the 10-pt staggered elliptic ladder family, which has R (L) loops to the right (left) of the rung with external legs p5 and p10, and M loops in between. The dashed lines ind…
Figure 3.1
Figure 3.1. Figure 3.1: (a) Representation of the map between dual-coordinate space xi and momentum-twistor space Zi , and (b) vice versa. Figure adapted from ref. [246]. where λi , λ˜ i are the spinor-helicity variables, defined via (pi) αα˙ ≡ p µ i (σµ) αα˙ = λ α i λ˜α˙ i , with σµ being …
Figure 3.2
Figure 3.2. Figure 3.2: Configuration of projective lines in momentum-twistor space as defined by the Schubert problem of the 1-loop 4-mass box integral. Parallel lines indicate that they do not intersect, although in general the lines are skew. Under the maximal cut, there are two solution…
Figure 3.3
Figure 3.3. Figure 3.3: Schubert problems for the 1-loop 4-mass box integrals {x1, x3, x5, x8} and {x1, x3, x6, x8}. For each Schubert problem, there are two lines (in blue), which simultaneously intersect all the respective external lines (in black) at the red dots. Moreover, the integrals…
Figure 3.4
Figure 3.4. Figure 3.4: Schubert problems for the 1-loop 4-mass box integrals {x1, x3, x5, x8}, {x1, x3, x6, x8} and {x1, x3, x8, x10}, which share three external lines. For each Schubert problem, there are two lines (in blue), which simultaneously intersect all the respective external line…
Figure 3.5
Figure 3.5. Figure 3.5: 2-loop Schubert problem for the 10-pt double-box integral. There are two lines (in blue) associated with the two loop momenta, which respectively intersect three external lines (in black) at the red dots. The two loop-momentum lines intersect at a point Zx that lies …
Figure 4.1
Figure 4.1. Figure 4.1: Four-loop PM diagrams depending on non-trivial geometries, organized by their self-force (SF) order, which will be defined in sec. 4.2.3. In total, there are two different CY threefolds (labeled by CY3 and CY′ 3 ) and two K3 surfaces (labeled by K3 and K3′ ). The geo…
Figure 4.2
Figure 4.2. Figure 4.2: Examples of diagrams with a power counting different than the classical one, or which do not contribute to the classical limit. (a) One-loop diagram with quantum power counting; (b) Superclassical one-loop box diagram; (c) and (d) One-loop diagrams which vanish in di…
Figure 4.3
Figure 4.3. Figure 4.3: Subgraphs which have zero master integrals. (a) One-loop bubble with at least one cubic matter vertex; (b) One-loop triangle with a cubic self-interaction and at least one cubic matter vertex at the matter line; (c) Two-loop dangling triangle with a quartic self-inte…
Figure 4.4
Figure 4.4. Figure 4.4: (a) Parametrization for the one-loop box; (b) Parametrization for the one-loop triangle. we have ∝ ∫ dz1dz2dz3dz4 z1z2z3z4 1 √ detG(k, u1, u2, q) . (4.43) In this expression, we drop the constant prefactors, and the Baikov polynomial takes the form detG(k, u1, u2, q)…
Figure 4.5
Figure 4.5. Figure 4.5: Parametrization for the two-loop double triangle. as can be seen from the parametrization, and as explained in sec. 4.3.2, we want to follow the loop-by-loop order {k1, k2}, as it avoids mixing k1 with q. Thus, starting with k1, we have a one-loop triangle such as th…
Figure 4.6
Figure 4.6. Figure 4.6: Parametrization for the diagram 3 of tab. 4.2. 4.6.1 A K3 surface at three loops In this subsection, we focus on the diagram in row 3 of tab. 4.2, and provide the details for the computation of the leading singularity for one integral of each parity. In particular, w…
Figure 4.7
Figure 4.7. Figure 4.7: Parametrization for the diagram 3 of tab. 4.3, which depends on a CY threefold in the odd-parity sector. 4.7.1 Non-trivial geometries I: Calabi–Yau threefolds In this subsection, we analyze in detail the 4-loop cases which depend on a Calabi–Yau threefold. In particu…
Figure 4.8
Figure 4.8. Figure 4.8: Parametrization for the diagram 37 of tab. 4.4, which depends on a CY threefold in the even-parity sector. Again, we have a degree-8 polynomial Q8(t1, t2, t3) in three variables, which is also quartic in each of them. By eqs. (1.13)–(1.17), it satisfies the Calabi–Ya…
Figure 4.9
Figure 4.9. Figure 4.9: Parametrizations for (a) the diagram 4 of tab. 4.3, and (b) the diagram 5 of tab. 4.3, which both depend on the 3-loop K3 surface in the odd-parity sector. 4.7.2 Non-trivial geometries II: K3 surfaces Having analyzed the two integrals which depend on three-dimensiona…
Figure 4.10
Figure 4.10. Figure 4.10: Parametrization for the diagram 6 of tab. 4.3, which depends on the 3-loop K3 surface in the even-parity sector. where the z14 that should appear due to the numerator factor has canceled against a similar factor in the denominator. Now, we can directly compute the r…
Figure 4.11
Figure 4.11. Figure 4.11: Parametrizations for (a) the diagram 38 of tab. 4.4, and (b) the diagram 39 of tab. 4.4, which both depend on the 3-loop K3 surface in the odd-parity sector. Upon noticing that the numerator and denominator in the second factor can be respectively written as 4z12 − …
Figure 4.12
Figure 4.12. Figure 4.12: Parametrization for the diagram 40 of tab. 4.4, which depends on a K3 surface (different than the 3-loop one) in the even-parity sector. parametrization given in fig. 4.12. With the loop-by-loop order {k1, k4, k3, k2}, and the 4 ISPs z11 = k 2 2 , z12 = (k2 − q) 2 ,…
Figure 5.1
Figure 5.1. Figure 5.1: Feynman integral topologies for the three subsectors (a)–(c) and the top sector (d) of the integral family in eq. (5.14). in fig. 4.8, and define the 11 propagators as D1 = k 2 1 , D2 = k 2 3 , D3 = (k4 + q) 2 , D4 = (k2 − q) 2 , (5.15a) D5 = (k4 − k3) 2 , D6 = (k1 −…
Figure 5.2
Figure 5.2. Figure 5.2: Different shapes of a self-dual ε-factorized differential equation matrix Aτ , and the resulting degree in ε for the apparent singularity of the corresponding Picard–Fuchs operator, up to 6 × 6 size. Thus, we now have an ansatz for the resulting ε-factorized differen…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

15 extracted references · 9 canonical work pages

  1. [290]

    Tsai, Weyl closure of a linear differential operator , Journal of Symbolic Computation 29 (2000) 747

    H. Tsai, Weyl closure of a linear differential operator , Journal of Symbolic Computation 29 (2000) 747

    work page 2000

  2. [291]

    Apparent Singularities of Linear Difference Equations with Polynomial Coefficients

    S.A. Abramov, M.A. Barkatou and M. Van Hoeij, Apparent singularities of linear differ- ence equations with polynomial coefficients , Appl. Algebra Eng., Commun. Comput. 17 (2006) 117–133 [math/0409508]

    work page Pith review arXiv 2006

  3. [292]

    S. Chen, M. Kauers and M.F. Singer, Desingularization of ore operators , Journal of Sym- bolic Computation 74 (2016) 617 [1408.5512]

    work page Pith review arXiv 2016

  4. [293]

    Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients

    S.Y. Slavyanov, D.A. Satco, A.M. Ishkhanyan and T.A. Rotinyan, Generation and re- moval of apparent singularities in linear ordinary differential equations with polynomial coefficients, Theor. Math. Phys. 189 (2016) 1726–1733 [1606.01476]

    work page Pith review arXiv 2016

  5. [294]

    S. Chen, M. Kauers, Z. Li and Y. Zhang, Apparent singularities of d-finite systems , J. Symb. Comput. 95 (2019) 217–237 [1705.00838]

    work page Pith review arXiv 2019

  6. [295]

    Kauers, M

    M. Kauers, M. Jaroschek and F. Johansson, Ore polynomials in sage, in Computer Algebra and Polynomials: Applications of Algebra and Number Theory , pp. 105–125, Springer International Publishing, 2015, DOI [1306.4263]

    arXiv 2015

  7. [296]

    Calabi-Yau operators of degree two

    G. Almkvist and D. van Straten, Calabi-Yau operators of degree two, J. Algebr. Comb. 58 (2023) 1203 [2103.08651]

    work page Pith review arXiv 2023

  8. [297]

    Candelas, X.C

    P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory , Nucl. Phys. B 359 (1991) 21

    work page 1991

Show all 15 references
  1. [298]

    Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, AMS/IP Stud

    D.R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, AMS/IP Stud. Adv. Math. 9 (1998) 185 [hep-th/9111025]

    1998 arXiv

  2. [299]

    van Straten, Calabi–Yau Operators, in Uniformization, Riemann-Hilbert Correspon- dence, Calabi-Yau Manifolds, and Picard-Fuchs Equations , pp

    D. van Straten, Calabi–Yau Operators, in Uniformization, Riemann-Hilbert Correspon- dence, Calabi-Yau Manifolds, and Picard-Fuchs Equations , pp. 401–451, Int. Press, 2017 [1704.00164]

    2017 arXiv

  3. [300]

    Almkvist, C

    G. Almkvist, C. van Enckevort, D. van Straten and W. Zudilin, Tables of Calabi–Yau equations, math/0507430

    arXiv

  4. [301]

    P¨ ogel, X

    S. P¨ ogel, X. Wang, S. Weinzierl, K. Wu and X. Xu, Self-dualities and Galois symmetries in Feynman integrals, JHEP 09 (2024) 084 [2407.08799]

    2024 arXiv

  5. [302]

    C. Duhr, F. Porkert, C. Semper and S.F. Stawinski, Self-duality from twisted cohomology, JHEP 03 (2025) 053 [2408.04904]

    2025

  6. [303]

    Remiddi and L

    E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph , Nucl. Phys. B 880 (2014) 343 [1311.3342]

    2014 arXiv

  7. [304]

    Tancredi, Integration by parts identities in integer numbers of dimensions

    L. Tancredi, Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations, Nucl. Phys. B 901 (2015) 282 [1509.03330]. 179

    2015 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.