Reversible linear differential equations

Malagón, Camilo Sanabria

doi:10.1016/j.jalgebra.2010.08.024

Cited by 2 publications

(5 citation statements)

References 16 publications

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“…A slightly more spectacular 23 example of order-8, G 6Dfcc 8 , has been found by Koutschan [27] for a six-dimensional face-centered cubic lattice. The irreducibility of this order-8 operator is hard to check 24 .…”

Section: Koutschan's Order-8 Operator: the Lattice Green Function Of ...mentioning

confidence: 77%

“…The exterior square of operator ( 23) is an irreducible order-5 operator (not order-6 as could be expected): one easily checks that the 'order-5 Calabi-Yau condition' (see [6] and (72) below) is actually satisfied for operator (23). If one considers an operator G4Dfcc 4 , non-trivially homomorphic [30,31] to G 4Dfcc 4 , its exterior square is, now, an operator of (the generic) order 6, and it has a rational solution.…”

Section: Special Lattice Green Odes: 4d Face-centered Cubic Latticementioning

confidence: 96%

“…In this case they can easily be recast into self-adjoint operators. A large set of linear differential operators conjugated to their adjoint, can be found in the very large list of Calabi-Yau order-4 operators obtained by Almkvist et al [5], or displayed by Batyrev and van Straten [4], or some simple order-3 operators displayed in a paper by Golyshev [21,22] (see also Sanabria Malagon [23]).…”

Section: Introductionmentioning

confidence: 99%

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Differential algebra on lattice Green functions and Calabi–Yau operators

Boukraa

Hassani

Maillard

et al. 2014

J. Phys. A: Math. Theor.

152

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We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups. We show that these irreducible operators are not only globally nilpotent, but are such that they are homomorphic to their (formal) adjoints. Considering these operators, or, sometimes, equivalent operators, we show that they are also such that, either their symmetric square or their exterior square, have a rational solution. This is a general result: an irreducible linear differential operator homomorphic to its (formal) adjoint is necessarily such that either its symmetric square, or its exterior square has a rational solution, and this situation corresponds to the occurrence of a special differential Galois group. We thus define the notion of being "Special Geometry" for a linear differential operator if it is irreducible, globally nilpotent, and such that it is homomorphic to its (formal) adjoint. Since many Derived From Geometry n-fold integrals ("Periods") occurring in physics, are seen to be diagonals of rational functions, we address several examples of (minimal order) operators annihilating diagonals of rational functions, and remark that they also seem to be, systematically, associated with irreducible factors homomorphic to their adjoint. -adjoint operators, homomorphism or equivalence of differential operators, Special Geometry, globally bounded series, diagonal of rational functions. † Sorbonne Universités (previously the UPMC was in Paris Universitas). ¶ Their corresponding linear differential operators are necessarily globally nilpotent [8]. ‡ One shows that there are no rational solutions of symmetric powers in degree 2, 3,4,6,8,9,12, using an algorithm in M. van Hoeij et al. [9] ♯ This operator is actually homomorphic to its adjoint (see below) with non-trivial order-two intertwiners. † This operator has an irregular singularity at infinity. At x = ∞ the solutions behave like: t · (1 + 77/72 t 2 + · · · ), and exp(−2/t)/t · (1 + 13/36 t + · · · ) where t = ±1/ √ x. † In maple the exterior power is normalised to be a monic operator (the head polynomial is normalised to 1). § The intertwiner T is given by the command Homomorphisms(L,L) of the DEtools package in Maple [33]. ♯ Note that the constraint on the order rules out the "tautological" intertwining relation, satisfied by any operator, like L · adjoint(T ) = T · adjoint(L) with T = L.¶ It is easy to show, in the case of an homomorphism of an operator L with its adjoint, that the intertwiner on the right-hand-side of (8) is necessarily equal to the adjoint of the intertwiner on the left-hand-side. Actually, from the equivalence L · T = S · adjoint(L), taking adjoint on both sides gives adjoint(T ) · adjoint(L) = L · adjoint(S). For irreducible L, the intertwiner is unique, so S = adjoint(T ). ¶ Corresponding to a change of variable: F (x) → F (x/(1 − 18 x))/(1 − 18 x).

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Section: Koutschan's Order-8 Operator: the Lattice Green Function Of ...mentioning

confidence: 77%

Section: Special Lattice Green Odes: 4d Face-centered Cubic Latticementioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Differential algebra on lattice Green functions and Calabi–Yau operators

Boukraa

Hassani

Maillard

et al. 2014

J. Phys. A: Math. Theor.

152

View full text Add to dashboard Cite

show abstract

“…The present paper is mainly concerned with (i). It clarifies and extends work of [Ber,S1,S2,S3]. Moreover we propose new methods and examples for (i) related to a theorem of Compoint [C, B] concerning invariants.…”

Section: Introductionmentioning

confidence: 56%

“…This method was refined in [vdP-U]. Klein's theorem is generalized in, e.g., [Ber,S1,S2,S3]. For (ii) there are many papers [S-U, Ho, H-W].…”

Section: Introductionmentioning

confidence: 99%

Linear differential equations with finite differential Galois group

2020

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For a differential operator L of order n over C(z) with a finite (differential) Galois group G ⊂ GL(C n ), there is an algorithm, by M. van Hoeij and J.-A. Weil, which computes the associated evaluation of the invariants ev :The procedure proposed here does the opposite: it uses a theorem of E. Compoint and computes the operator L from a given evaluation h. Moreover it solves a part of the inverse problem of producing L for a given representation of a finite group G. Another part considered here, is finding irreducible G-invariant curves Z ⊂ P(C n ) with Z/G of genus zero and constructing evaluations from this. The theory developed here is illustrated by various examples, and relates to and continues classical work of H.

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Reversible linear differential equations

Cited by 2 publications

References 16 publications

Differential algebra on lattice Green functions and Calabi–Yau operators

Differential algebra on lattice Green functions and Calabi–Yau operators

Linear differential equations with finite differential Galois group

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