Computing closed form solutions of integrable connections

Barkatou, Moulay A.; Cluzeau, Thomas; Bacha, Carole El; Weil, Jacques-Arthur

doi:10.1145/2442829.2442840

Cited by 20 publications

(48 citation statements)

References 13 publications

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“…If one assumes that the Wronskian of the order-four linear differential operator is a square of a rational function, one, thus, finds a rational solution for the exterior square of the differential system (resp. hyperexponential solution [42] for a Wronskian N -th root of a rational function).…”

Section: Calabi-yau Conditions and Rational Solutions Of The Exteriormentioning

confidence: 99%

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: Calabi-yau Conditions and Rational Solutions Of The Exteriormentioning

confidence: 99%

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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“…This notion, in the univariate case, gives another approach to construct a basis of the space of regular solutions [12], and the obstacles encountered are similar to those in rank reduction. An additional field is to study closed form solutions [10] in the light of associated ODS introduced in Section 4.…”

Section: Resultsmentioning

confidence: 99%

“…The polynomial det(G Ai (λ)) vanishes identically in λ if and only if θ i (λ) given by (10) does. In fact, let D(x i ) = Diag(x i I r , I d−r ).…”

Section: Converse Of Theorem 18mentioning

confidence: 99%

Formal solutions of completely integrable Pfaffian systems with normal crossings

Maddah

2015

ACM Commun. Comput. Algebra

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“…Hence, by (11) and [30, Theorem 1], θ i (λ) = 0. ✷ Intuitively, the characteristic polynomial of A i /x i is used in (10) to detect the true Poincaré rank of the i th component via the valuation of x i . It turns out that the valuation is only influenced by A i,0 and A i,1 .…”

Section: Main Theoremsmentioning

confidence: 99%