Galois Theories of Linear Difference
                    Equations: An Introduction

Hardouin, Charlotte; Sauloy, Jacques; Singer, Michael F.

doi:10.1090/surv/211

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“…The main result will be that if such a q-difference equation admits a formal limit when q → 1, then it admits a fundamental solution whose limit is a fundamental solution of the limit differential equation. The main references for this section are [Sau00] and [HSS16].…”

Section: V2 Survey Of the Theory Of Regular Singular Q-difference Equ...mentioning

confidence: 99%

“…Remark V.2.1.12. In [HSS16], such a q-difference system is called fuchsian at 0. A q-difference system is strictly fuchsian at 0 if its associated matrix is already invertible without the need of a q-gauge transform.…”

Section: V21 Fundamental Solution Q-gauge Transform and Q-pullbackmentioning

confidence: 99%

“…We start by recursively building a gauge transform F q ∈ GL n C Q, Q −1 sending the matrix A q (Q) to the constant matrix A q (0). See [Sau00], Subsections 1.1.1 and 1.1.3, or [HSS16], Theorems 3.2.2 and 3.2.3 pp.127-129 for the construction.…”

Section: Chapter V Regular Singular Q-difference Equationsmentioning

confidence: 99%

“…The data of Birkhoff's connection matrix classifies fuchsian q-difference systems up to gauge transform. For a precise statement, see [HSS16], Theorem 3.4.9 p.134.…”

Section: V3 Monodromy Of Regular Singular Q-difference Equationsmentioning

confidence: 99%

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Confluence of quantum $K$-theory to quantum cohomology for projective spaces

Roquefeuil

2019

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This thesis was financed by Université d'Angers (imputation budgétaire A900210).The author also received financial support from from Agence Nationale de la Recherche's projects ANR-13-IS01-0001 (SISYPH) and ANR-17-CE40-0014 (CatAG). CHAPTER I. INTRODUCTIONTo answer the second question (Q2), there are two approaches. The first approach is to look for an application of a Riemann-Roch theorem. Unfortunately, a Hirzebruch-Riemann-Roch formula we could use for Deligne-Mumford stack, due to B. Toën [Toe99], only works when these stacks are smooth -which our moduli spaces are not in general (though this is the case when the target space X is a projective space). In 2011, A. Givental and V. Tonita [GT14] prove a theorem relating the two Gromov-Witten theories. The statements are nonetheless quite technical.The second approach is the aim of this thesis. It is inspired by the computations of the K-theoretical J-function by Givental-Lee [GL03]. In the case X = P N , the J-function can be linked to the q-hypergeometric seriesGiven a q-difference equation, we are able to obtain a differential equation using a phenomenon called confluence. We can therefore wonder if we can compare the confluence of q-difference equations in quantum K-theory with the differential equations in quantum cohomology. Confluence and Gromov-Witten theoriesThe confluence phenomenon for q-difference equations was studied first by J. Sauloy in 2000 [Sau00]. This property says that a q-difference equation can admit a differential equation as a limit by doing "q → 1". Notice that for any k ∈ Z we haveBecause of this, we have the formal limit limThis principle generalises to general q-difference equations, as long as we allow ourselves to specify further what "q → 1" means. A q-difference equation that has a well defined limit when q tends to 1 is said to be confluent (see Definition V.2.4.4). Then, the limit of this q-difference equation defines a differential equation. It makes sense to compare the q-difference equation and its limit: for example, the solutions of the q-difference equation give solutions to the differential equation by taking their limit when q tends to 1 (see Theorem V.2.4.7).The main result of this thesis adapts the confluence of q-difference equation in the context of quantum K-theory to obtain the theorem below.Theorem (VI.2.1.1). For X = P N , let J coh (resp. J Kth ) be the small cohomological (resp. K-theoretical) J-function. Then, 1. The q-difference equation satisfied by J Kth will degenerate through confluence to the differential equation satisfied by J coh .2. We denote by ch ∶ K P N ⊗ Q → H * P N ; Q the Chern character. Let confluence J Kth the result of confluence applied to the solution J Kth . Then, we have ch confluence J Kth = J coh I.2. PLAN OF THE THESIS 9Chapter II Moduli space of stable mapsIn this introductory chapter we review the construction of the moduli space of stable maps, which is an essential ingredient to the define Gromov-Witten invariants. Then, we define the evaluation maps and the tautological cotangen...

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Section: V2 Survey Of the Theory Of Regular Singular Q-difference Equ...mentioning

confidence: 99%

Section: V21 Fundamental Solution Q-gauge Transform and Q-pullbackmentioning

confidence: 99%

Section: Chapter V Regular Singular Q-difference Equationsmentioning

confidence: 99%

“…The data of Birkhoff's connection matrix classifies fuchsian q-difference systems up to gauge transform. For a precise statement, see [HSS16], Theorem 3.4.9 p.134.…”

Section: V3 Monodromy Of Regular Singular Q-difference Equationsmentioning

confidence: 99%

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Confluence of quantum $K$-theory to quantum cohomology for projective spaces

Roquefeuil

2019

Preprint

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“…By [21,Prop. 6.24] and [18,Rem. 4.37], R is the coordinate ring of a G-torsor over k, which implies that there is a δ-field extensionk of k and a G-equivariant isomorphism (wherek is endowed with the trivial G-action) ofk-algebrask ⊗ k R ≃k ⊗ C C{G}, where C{G} denotes the δ-Hopf algebra of coordinate functions on G [34,Def.…”

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confidence: 99%

Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

Arreche

2017

Commun. Contemp. Math.

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We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation of the form y(x + 2) + a(x)y(x + 1) + b(x)y(x) = 0, where the coefficients a(x), b(x) ∈Q(x) are rational functions in x with coefficients inQ. We develop algorithms to compute the differencedifferential Galois group associated to such an equation, and show how to deduce the differentialalgebraic relations among the solutions from the defining equations of the Galois group. IntroductionConsider a second-order homogeneous linear difference equation σ 2 (y) + aσ(y) + by = 0, (1.1) whose coefficients a, b ∈Q(x), and where σ denotes theQ-linear automorphism defined by σ(x) = x + 1. We are motivated by the question: do the solutions of (1.1) satisfy any d dx -algebraic equations overQ(x)? And if so, how can we compute all such differential-algebraic relations? We give complete answers to these questions as an application of the difference-differential Galois theory developed in [21], which studies equations such as (1.1) from a purely algebraic point of view. This theory attaches a linear differential algebraic group G (Definition 2.7) to (1.1), which group encodes all the difference-differential algebraic relations among the solutions to (1.1). We develop an algorithm to compute G, and then show how the knowledge of G leads to a concrete description of the sought difference-differential algebraic relations among the solutions. The difference-differential Galois theory of [21] is a generalization of the difference Galois theory presented in [38], where the Galois groups that arise are linear algebraic groups that encode the difference-algebraic relations among the solutions to a given linear difference equation. An algorithm to compute the difference Galois group H associated to (1.1) by the theory of [38] is developed in [23]. The computation of G is more difficult than that of H, because there are many more linear differential algebraic groups than there are linear algebraic groups (more precisely, the latter are instances of the former), so identifying the correct differencedifferential Galois group from among these possibilities requires additional work. However, the difference Galois group H serves as a close upper bound for the difference-differential Galois group G: it is shown in [21] that one can consider G as a Zariski-dense subgroup of H without loss of generality (see Proposition 2.12 for a precise statement). In view of this fact, our strategy to compute G is to first apply the algorithm of [23] to compute H, and then compute the additional differential-algebraic equations (if any) that define G as a subgroup of H.This strategy is reminiscent of the one begun in [11], and concluded in [2][3][4], to compute the parameterized differential Galois group for a second-order linear differential equation with differential parameters, where the results of [6,28] are first applied to compute the classical (non-...

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Difference Galois Theory for the “Applied” Mathematician

Vizio¹

2020

Lecture Notes in Mathematics

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Galois Theories of Linear Difference Equations: An Introduction

Cited by 13 publications

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Confluence of quantum $K$-theory to quantum cohomology for projective spaces

Confluence of quantum $K$-theory to quantum cohomology for projective spaces

Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

Difference Galois Theory for the “Applied” Mathematician

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