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Introduction 2-descent C2× C2-descent Tables Methods
Solve Linear Differential Equations in terms ofHypergeometric Functions
Tingting Fang & Mark van HoeijFlorida State University
Boston, USA
January 6th, 2012
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 1/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Differential operator and differential equation
LetL = an∂
n + an−1∂n−1 + · · ·+ a1∂ + a0
be a differential operator, with an, an−1, · · · , a1, a0 ∈ C(x) and npositive integer. The corresponding differential equation is
any (n) + an−1y (n−1) + · · ·+ a1y ′ + a0y = 0.
Note: we are interested in a class of equations that can be solvedin terms of Hypergeometric Functions.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 2/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Differential operator and differential equation
LetL = an∂
n + an−1∂n−1 + · · ·+ a1∂ + a0
be a differential operator, with an, an−1, · · · , a1, a0 ∈ C(x) and npositive integer. The corresponding differential equation is
any (n) + an−1y (n−1) + · · ·+ a1y ′ + a0y = 0.
Note: we are interested in a class of equations that can be solvedin terms of Hypergeometric Functions.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 2/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Traditional Methods of Solving Differential Operator L
Direct solving by the existing techniques.
Factor L as a product of lower order differential operators,then solve L by solving the lower order ones.
Solve L in terms of lower order differential operator.
My talk here focuses on second order linear differential equations(differential operators) which are irreducible and have noLiouvillian solutions.Question: For the equations that we can’t solve by the abovetechniques, what should we do?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 3/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Traditional Methods of Solving Differential Operator L
Direct solving by the existing techniques.
Factor L as a product of lower order differential operators,then solve L by solving the lower order ones.
Solve L in terms of lower order differential operator.
My talk here focuses on second order linear differential equations(differential operators) which are irreducible and have noLiouvillian solutions.Question: For the equations that we can’t solve by the abovetechniques, what should we do?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 3/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Traditional Methods of Solving Differential Operator L
Direct solving by the existing techniques.
Factor L as a product of lower order differential operators,then solve L by solving the lower order ones.
Solve L in terms of lower order differential operator.
My talk here focuses on second order linear differential equations(differential operators) which are irreducible and have noLiouvillian solutions.Question: For the equations that we can’t solve by the abovetechniques, what should we do?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 3/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Traditional Methods of Solving Differential Operator L
Direct solving by the existing techniques.
Factor L as a product of lower order differential operators,then solve L by solving the lower order ones.
Solve L in terms of lower order differential operator.
My talk here focuses on second order linear differential equations(differential operators) which are irreducible and have noLiouvillian solutions.Question: For the equations that we can’t solve by the abovetechniques, what should we do?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 3/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Traditional Methods of Solving Differential Operator L
Direct solving by the existing techniques.
Factor L as a product of lower order differential operators,then solve L by solving the lower order ones.
Solve L in terms of lower order differential operator.
My talk here focuses on second order linear differential equations(differential operators) which are irreducible and have noLiouvillian solutions.
Question: For the equations that we can’t solve by the abovetechniques, what should we do?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 3/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Traditional Methods of Solving Differential Operator L
Direct solving by the existing techniques.
Factor L as a product of lower order differential operators,then solve L by solving the lower order ones.
Solve L in terms of lower order differential operator.
My talk here focuses on second order linear differential equations(differential operators) which are irreducible and have noLiouvillian solutions.Question: For the equations that we can’t solve by the abovetechniques, what should we do?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 3/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Overview of the methods
We consider to reduce the differential operator L, if possible, toanother differential operator L that is easier to solve (with sameorder, but with fewer true singularities) by using the 2-descentmethod or other descent methods.
1 If the above 2-descent exists, we find L.
2 If the number of true singularities of L drops to 3, we find its
2F1-type solutions, furthermore, find the 2F1 solution of L interms of L’s.
3 If the number of true singularities of L drops to 4, we candecide if L, furthermore L, ∃ 2F1-type solutions by building alarge table that covers the differential operators with 4 truesingularities.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 4/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Overview of the methods
We consider to reduce the differential operator L, if possible, toanother differential operator L that is easier to solve (with sameorder, but with fewer true singularities) by using the 2-descentmethod or other descent methods.
1 If the above 2-descent exists, we find L.
2 If the number of true singularities of L drops to 3, we find its
2F1-type solutions, furthermore, find the 2F1 solution of L interms of L’s.
3 If the number of true singularities of L drops to 4, we candecide if L, furthermore L, ∃ 2F1-type solutions by building alarge table that covers the differential operators with 4 truesingularities.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 4/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Overview of the methods
We consider to reduce the differential operator L, if possible, toanother differential operator L that is easier to solve (with sameorder, but with fewer true singularities) by using the 2-descentmethod or other descent methods.
1 If the above 2-descent exists, we find L.
2 If the number of true singularities of L drops to 3, we find its
2F1-type solutions, furthermore, find the 2F1 solution of L interms of L’s.
3 If the number of true singularities of L drops to 4, we candecide if L, furthermore L, ∃ 2F1-type solutions by building alarge table that covers the differential operators with 4 truesingularities.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 4/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Overview of the methods
We consider to reduce the differential operator L, if possible, toanother differential operator L that is easier to solve (with sameorder, but with fewer true singularities) by using the 2-descentmethod or other descent methods.
1 If the above 2-descent exists, we find L.
2 If the number of true singularities of L drops to 3, we find its
2F1-type solutions, furthermore, find the 2F1 solution of L interms of L’s.
3 If the number of true singularities of L drops to 4, we candecide if L, furthermore L, ∃ 2F1-type solutions by building alarge table that covers the differential operators with 4 truesingularities.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 4/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Transformations
When we talk about that L can be solved in terms of the solutionsof L, we mean that L can be transformed to L.
There are three types of transformations that preserve order 2:
1 change of variables: y(x)→ y(f (x)), f (x) ∈ C(x) \ C.
2 exp-product: y → e∫r dx · y , r ∈ C(x).
3 gauge transformation: y → r0y + r1y ′, r0, r1 ∈ C(x).
Given L1, L2 ∈ C(x)[∂] with order 2:
If L12&3→ L2, then L1 ∼p L2 (projectively equivalent)
If L13→ L2, then L1 ∼g L2 (gauge equivalent).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 5/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Transformations
When we talk about that L can be solved in terms of the solutionsof L, we mean that L can be transformed to L.There are three types of transformations that preserve order 2:
1 change of variables: y(x)→ y(f (x)), f (x) ∈ C(x) \ C.
2 exp-product: y → e∫r dx · y , r ∈ C(x).
3 gauge transformation: y → r0y + r1y ′, r0, r1 ∈ C(x).
Given L1, L2 ∈ C(x)[∂] with order 2:
If L12&3→ L2, then L1 ∼p L2 (projectively equivalent)
If L13→ L2, then L1 ∼g L2 (gauge equivalent).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 5/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Transformations
When we talk about that L can be solved in terms of the solutionsof L, we mean that L can be transformed to L.There are three types of transformations that preserve order 2:
1 change of variables: y(x)→ y(f (x)), f (x) ∈ C(x) \ C.
2 exp-product: y → e∫r dx · y , r ∈ C(x).
3 gauge transformation: y → r0y + r1y ′, r0, r1 ∈ C(x).
Given L1, L2 ∈ C(x)[∂] with order 2:
If L12&3→ L2, then L1 ∼p L2 (projectively equivalent)
If L13→ L2, then L1 ∼g L2 (gauge equivalent).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 5/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Introduction
Transformations
When we talk about that L can be solved in terms of the solutionsof L, we mean that L can be transformed to L.There are three types of transformations that preserve order 2:
1 change of variables: y(x)→ y(f (x)), f (x) ∈ C(x) \ C.
2 exp-product: y → e∫r dx · y , r ∈ C(x).
3 gauge transformation: y → r0y + r1y ′, r0, r1 ∈ C(x).
Given L1, L2 ∈ C(x)[∂] with order 2:
If L12&3→ L2, then L1 ∼p L2 (projectively equivalent)
If L13→ L2, then L1 ∼g L2 (gauge equivalent).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 5/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Example 1
L = x2(36x2 − 1)(4x2 − 1)(12x2 − 1)∂2+
4x(2x − 1)(1296x5 + 576x4 − 144x3 − 72x2 + x + 1)∂+
2(5184x6 − 864x5 − 1656x4 + 48x3 + 162x2 + 6x − 1)
Question: How to find the 2F1 hypergeometric solution of L?
y1 =r1 · 2F1
(1/4, 1/4
3/2144x4+24x2+1
64x2
)+r2 · 2F1
(5/4, 5/4
5/2144x4+24x2+1
64x2
)(with r1, r2 ∈ C(x))
y2 = · · ·
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 6/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Example 1
L = x2(36x2 − 1)(4x2 − 1)(12x2 − 1)∂2+
4x(2x − 1)(1296x5 + 576x4 − 144x3 − 72x2 + x + 1)∂+
2(5184x6 − 864x5 − 1656x4 + 48x3 + 162x2 + 6x − 1)
Question: How to find the 2F1 hypergeometric solution of L?
y1 =r1 · 2F1
(1/4, 1/4
3/2144x4+24x2+1
64x2
)+r2 · 2F1
(5/4, 5/4
5/2144x4+24x2+1
64x2
)(with r1, r2 ∈ C(x))
y2 = · · ·
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 6/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Example 1
L = x2(36x2 − 1)(4x2 − 1)(12x2 − 1)∂2+
4x(2x − 1)(1296x5 + 576x4 − 144x3 − 72x2 + x + 1)∂+
2(5184x6 − 864x5 − 1656x4 + 48x3 + 162x2 + 6x − 1)
Question: How to find the 2F1 hypergeometric solution of L?
y1 =r1 · 2F1
(1/4, 1/4
3/2144x4+24x2+1
64x2
)+r2 · 2F1
(5/4, 5/4
5/2144x4+24x2+1
64x2
)(with r1, r2 ∈ C(x))
y2 = · · ·
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 6/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Definition for 2-descent
Given a second order differential operator L defined over C(x), wesay that L has 2-descent if ∃f ∈ C(x) with degree(f )= 2, and∃L ∈ C(f )[∂f ] such that L ∼p L.
Note: ∂f = ddf = 1
f ′∂
Two steps to find the 2-descent L
1 Finding the subfield C(f ) with [C(x) : C(f )] = 2, i.e. findingf ∈ C(x) of degree 2.
2 Finding the projectively equivalent differential operatorL ∈ C(f )[∂f ].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 7/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Definition for 2-descent
Given a second order differential operator L defined over C(x), wesay that L has 2-descent if ∃f ∈ C(x) with degree(f )= 2, and∃L ∈ C(f )[∂f ] such that L ∼p L.Note: ∂f = d
df = 1f ′∂
Two steps to find the 2-descent L
1 Finding the subfield C(f ) with [C(x) : C(f )] = 2, i.e. findingf ∈ C(x) of degree 2.
2 Finding the projectively equivalent differential operatorL ∈ C(f )[∂f ].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 7/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Definition for 2-descent
Given a second order differential operator L defined over C(x), wesay that L has 2-descent if ∃f ∈ C(x) with degree(f )= 2, and∃L ∈ C(f )[∂f ] such that L ∼p L.Note: ∂f = d
df = 1f ′∂
Two steps to find the 2-descent L
1 Finding the subfield C(f ) with [C(x) : C(f )] = 2, i.e. findingf ∈ C(x) of degree 2.
2 Finding the projectively equivalent differential operatorL ∈ C(f )[∂f ].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 7/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Definition for 2-descent
Given a second order differential operator L defined over C(x), wesay that L has 2-descent if ∃f ∈ C(x) with degree(f )= 2, and∃L ∈ C(f )[∂f ] such that L ∼p L.Note: ∂f = d
df = 1f ′∂
Two steps to find the 2-descent L
1 Finding the subfield C(f ) with [C(x) : C(f )] = 2, i.e. findingf ∈ C(x) of degree 2.
2 Finding the projectively equivalent differential operatorL ∈ C(f )[∂f ].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 7/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Definition for 2-descent
Given a second order differential operator L defined over C(x), wesay that L has 2-descent if ∃f ∈ C(x) with degree(f )= 2, and∃L ∈ C(f )[∂f ] such that L ∼p L.Note: ∂f = d
df = 1f ′∂
Two steps to find the 2-descent L
1 Finding the subfield C(f ) with [C(x) : C(f )] = 2, i.e. findingf ∈ C(x) of degree 2.
2 Finding the projectively equivalent differential operatorL ∈ C(f )[∂f ].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 7/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Find the subfield L descending to
Since every extension of degree 2 is Galois, so by Luroth’s theorm,we have the following relationship:
Remark
A subfield C(f ) ⊂ C(x) with [C(x) : C(f )] = 2
⇐⇒
σ ∈ Aut(C(x)/C) with degree 2
Necessary Requirements for σ
σ = ax+bcx+d with d = −a;
σ should preserve the set of true singularities of L and theirexponent-difference mod Z.
For each such σ, we compute a candidate subfield C(f ) ⊆ C(x).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 8/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Find the subfield L descending to
Since every extension of degree 2 is Galois, so by Luroth’s theorm,we have the following relationship:
Remark
A subfield C(f ) ⊂ C(x) with [C(x) : C(f )] = 2
⇐⇒
σ ∈ Aut(C(x)/C) with degree 2
Necessary Requirements for σ
σ = ax+bcx+d with d = −a;
σ should preserve the set of true singularities of L and theirexponent-difference mod Z.
For each such σ, we compute a candidate subfield C(f ) ⊆ C(x).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 8/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Find the subfield L descending to
Since every extension of degree 2 is Galois, so by Luroth’s theorm,we have the following relationship:
Remark
A subfield C(f ) ⊂ C(x) with [C(x) : C(f )] = 2
⇐⇒
σ ∈ Aut(C(x)/C) with degree 2
Necessary Requirements for σ
σ = ax+bcx+d with d = −a;
σ should preserve the set of true singularities of L and theirexponent-difference mod Z.
For each such σ, we compute a candidate subfield C(f ) ⊆ C(x).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 8/ 23
Introduction 2-descent C2× C2-descent Tables Methods
2-descent Method
Find the subfield L descending to
Since every extension of degree 2 is Galois, so by Luroth’s theorm,we have the following relationship:
Remark
A subfield C(f ) ⊂ C(x) with [C(x) : C(f )] = 2
⇐⇒
σ ∈ Aut(C(x)/C) with degree 2
Necessary Requirements for σ
σ = ax+bcx+d with d = −a;
σ should preserve the set of true singularities of L and theirexponent-difference mod Z.
For each such σ, we compute a candidate subfield C(f ) ⊆ C(x).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 8/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Theoretical support
The following σ and C(f ) represent the Mobius transformationfound previously and the corresponding fixed field, respectively.Suppose L descends to L ∈ C(f )[∂f ], we have
L ∼p L = σ(L) ∼p σ(L), and so L ∼p σ(L)
which means we can find the projective equivalence:
y → e∫r dx · (r0y + r1y ′)
from the solution space of L to the solution space of σ(L).Question: How to compute L from it?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 9/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Theoretical support
The following σ and C(f ) represent the Mobius transformationfound previously and the corresponding fixed field, respectively.Suppose L descends to L ∈ C(f )[∂f ], we have
L ∼p L = σ(L) ∼p σ(L), and so L ∼p σ(L)
which means we can find the projective equivalence:
y → e∫r dx · (r0y + r1y ′)
from the solution space of L to the solution space of σ(L).
Question: How to compute L from it?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 9/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Theoretical support
The following σ and C(f ) represent the Mobius transformationfound previously and the corresponding fixed field, respectively.Suppose L descends to L ∈ C(f )[∂f ], we have
L ∼p L = σ(L) ∼p σ(L), and so L ∼p σ(L)
which means we can find the projective equivalence:
y → e∫r dx · (r0y + r1y ′)
from the solution space of L to the solution space of σ(L).Question: How to compute L from it?
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 9/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case A
Case A is when L ∼g σ(L), in other words, there exists G = r0+r1 ∂ ∈ C(x)[∂] with G (V (L)) = V (σ(L)). Then ∃ L ∈ C(f )[∂f ]with L ∼g L.
Question: Given G , how to find L?
V (L)
V (L)
A
-
G -
=
V (σ(L))
V (σ(L))�
σ(A)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 10/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case A
Case A is when L ∼g σ(L), in other words, there exists G = r0+r1 ∂ ∈ C(x)[∂] with G (V (L)) = V (σ(L)). Then ∃ L ∈ C(f )[∂f ]with L ∼g L.Question: Given G , how to find L?
V (L)
V (L)
A
-
G -
=
V (σ(L))
V (σ(L))�
σ(A)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 10/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case A
Case A is when L ∼g σ(L), in other words, there exists G = r0+r1 ∂ ∈ C(x)[∂] with G (V (L)) = V (σ(L)). Then ∃ L ∈ C(f )[∂f ]with L ∼g L.Question: Given G , how to find L?
V (L)
V (L)
A
-
G -
=
V (σ(L))
V (σ(L))�
σ(A)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 10/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case A
Case A is when L ∼g σ(L), in other words, there exists G = r0+r1 ∂ ∈ C(x)[∂] with G (V (L)) = V (σ(L)). Then ∃ L ∈ C(f )[∂f ]with L ∼g L.Question: Given G , how to find L?
V (L)
V (L)
A
-
G -
=
V (σ(L))
V (σ(L))�
σ(A)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 10/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case A
Case A is when L ∼g σ(L), in other words, there exists G = r0+r1 ∂ ∈ C(x)[∂] with G (V (L)) = V (σ(L)). Then ∃ L ∈ C(f )[∂f ]with L ∼g L.Question: Given G , how to find L?
V (L)
V (L)
A
-
G -
=
V (σ(L))
V (σ(L))�
σ(A)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 10/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Question arising in the above diagram
Question: When does the above diagram commute?
Theorem
Let L and σ be as before, and G : V (L)→ V (σ(L)) be a gaugetransformation. Suppose L1, L2 ∈ C(f )[∂f ] and Ai : V (L)→ V (Li )are gauge transformations. Then:
1 For each i = 1, 2, there is exactly one λi ∈ C∗ such thatthe following diagram commutes
2 If L1 ∼g L2 over C(f ), then λ1 = λ2; Otherwise, λ1 = −λ2.
3 In particular, {λ1,−λ1} depends only on (L, σ,G ).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 11/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Question arising in the above diagram
Question: When does the above diagram commute?
Theorem
Let L and σ be as before, and G : V (L)→ V (σ(L)) be a gaugetransformation. Suppose L1, L2 ∈ C(f )[∂f ] and Ai : V (L)→ V (Li )are gauge transformations. Then:
1 For each i = 1, 2, there is exactly one λi ∈ C∗ such thatthe following diagram commutes
2 If L1 ∼g L2 over C(f ), then λ1 = λ2; Otherwise, λ1 = −λ2.
3 In particular, {λ1,−λ1} depends only on (L, σ,G ).
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 11/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Diagram
V (L)λi G - V (σ(L))
V (Li )�
σ(A i)
Ai
-
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 12/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case B
Case B is when L ∼p σ(L), in other words, there exists G = e∫r ·
(r0 + r1 ∂) such that G (V (L)) = V (σ(L)).
Difficulty
We have an exponential part in G comparing with Case A. Thealgorithm mentioned above fails.
Solution
After multiplying solution of L by a suitable e∫s , we can reduce
this case to Case A.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 13/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case B
Case B is when L ∼p σ(L), in other words, there exists G = e∫r ·
(r0 + r1 ∂) such that G (V (L)) = V (σ(L)).
Difficulty
We have an exponential part in G comparing with Case A. Thealgorithm mentioned above fails.
Solution
After multiplying solution of L by a suitable e∫s , we can reduce
this case to Case A.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 13/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Case B
Case B is when L ∼p σ(L), in other words, there exists G = e∫r ·
(r0 + r1 ∂) such that G (V (L)) = V (σ(L)).
Difficulty
We have an exponential part in G comparing with Case A. Thealgorithm mentioned above fails.
Solution
After multiplying solution of L by a suitable e∫s , we can reduce
this case to Case A.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 13/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Finding the projectively equivalent operator L
Application for this result, Example 2
Start with L = ∂2 + 8(8x+1)(4x+1)(4x−1)∂ + 4(8x+1)
x(4x−1)(4x+1) ;
End with 2-descent L = (16x1 − 1)x1∂2 + (32x1 − 1)∂ + 4.
Solution of L :
y1 =2F1
(1/2, 1/2
1−16x + 1
); y2 = 2F1
(1/2, 1/2
116x
)Solution of L :
y1 =1
2· 2F1
(1/2, 1/2
2−16x2 + 1
)−
x(4x + 1)
2· 2F1
(3/2, 3/2
3−16x2 + 1
)
y2 =4x
4x − 1· 2F1
(1/2, 1/2
116x2
)+
8x2(4x + 1)
4x − 1· 2F1
(3/2, 3/2
216x2
)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 14/ 23
Introduction 2-descent C2× C2-descent Tables Methods
C2× C2-descent
C2× C2-descent
C(x)
C(f1)�
σ 1
C(f2)
σ2
?C(f3)
σ3
-
C(f )?�
-
[C(x) : C(f )] = 4 and the Galois group is C 2× C 2.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 15/ 23
Introduction 2-descent C2× C2-descent Tables Methods
C2× C2-descent
C2× C2-descent
C(x)
C(f1)�
σ 1
C(f2)
σ2
?C(f3)
σ3
-
C(f )?�
-
[C(x) : C(f )] = 4 and the Galois group is C 2× C 2.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 15/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Solve L with 4 true singularities by constructing tables
Main Idea
After implementing 2-decent,C 2× C 2-descent or some otherdescent methods, We may end up with L with 4 true singularities,not 3 true singularities. This time we consider to solve such L byconstructing tables.
Given L with 4 true singularities, we are trying to find the followingform of solutions of L:
e∫r · (r0y + r1y
′)
here r , r0, r1 ∈ C(x), and y = 2F1(a, b; c |f ).Main idea: Given L with 4 true singularities, we first figure outthe ramification type of the singularities, and then match this withthe table we have constructed. If we have the matched ones, wefind the corresponding f in the table. For the left exponential partand the r0 and r1, we implement the equivalence programs to findthem [9].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 16/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Solve L with 4 true singularities by constructing tables
Main Idea
After implementing 2-decent,C 2× C 2-descent or some otherdescent methods, We may end up with L with 4 true singularities,not 3 true singularities. This time we consider to solve such L byconstructing tables.Given L with 4 true singularities, we are trying to find the followingform of solutions of L:
e∫r · (r0y + r1y
′)
here r , r0, r1 ∈ C(x), and y = 2F1(a, b; c |f ).
Main idea: Given L with 4 true singularities, we first figure outthe ramification type of the singularities, and then match this withthe table we have constructed. If we have the matched ones, wefind the corresponding f in the table. For the left exponential partand the r0 and r1, we implement the equivalence programs to findthem [9].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 16/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Solve L with 4 true singularities by constructing tables
Main Idea
After implementing 2-decent,C 2× C 2-descent or some otherdescent methods, We may end up with L with 4 true singularities,not 3 true singularities. This time we consider to solve such L byconstructing tables.Given L with 4 true singularities, we are trying to find the followingform of solutions of L:
e∫r · (r0y + r1y
′)
here r , r0, r1 ∈ C(x), and y = 2F1(a, b; c |f ).Main idea: Given L with 4 true singularities, we first figure outthe ramification type of the singularities, and then match this withthe table we have constructed. If we have the matched ones, wefind the corresponding f in the table. For the left exponential partand the r0 and r1, we implement the equivalence programs to findthem [9].
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 16/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Solve L with 4 true singularities by constructing tables
Riemann-Hurwitz Theorem
Question: How do we construct these f in the table?
By Riemann-Hurwitz Theorem, the ramification type of thesingularities should satisfy the equation
2n − 2 =∑p∈P1
(ep − 1)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 17/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Solve L with 4 true singularities by constructing tables
Riemann-Hurwitz Theorem
Question: How do we construct these f in the table?By Riemann-Hurwitz Theorem, the ramification type of thesingularities should satisfy the equation
2n − 2 =∑p∈P1
(ep − 1)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 17/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Belyi Maps
Table for Belyi Maps
When f is a Belyi Map, that means all the ramification points p off occur over the set {0, 1,∞}. This part of classification of f s isdone by Dr. van Hoeij and Dr. Vidunas [7].
Degree bound for f Number of Cases
Non Parametric Case 60 366
Parametric Case 12 48
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 18/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Belyi Maps
Table for Belyi Maps
When f is a Belyi Map, that means all the ramification points p off occur over the set {0, 1,∞}. This part of classification of f s isdone by Dr. van Hoeij and Dr. Vidunas [7].
Degree bound for f Number of Cases
Non Parametric Case 60 366
Parametric Case 12 48
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 18/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Near Belyi Maps
Table for Near Belyi Maps
When f is a near Belyi Map, we mean that f ramifies only above 4points, and such that there is a simple ramified point in one ofthese fibers. (The other ramified points occur over {0, 1,∞})
Degree bound for f Number of Cases
Non Parametric Case 18 ≥ 10
Parametric Case 6 ≥ 7
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 19/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Near Belyi Maps
Table for Near Belyi Maps
When f is a near Belyi Map, we mean that f ramifies only above 4points, and such that there is a simple ramified point in one ofthese fibers. (The other ramified points occur over {0, 1,∞})
Degree bound for f Number of Cases
Non Parametric Case 18 ≥ 10
Parametric Case 6 ≥ 7
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 19/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Near Belyi Maps
Parametric Case for near Belyi Maps
Exponent-Difference Ramification type Deg ftype (Pattern)
(1/2, a, b) (2),(1,1),(1,1) 2 Has 2-descent
(1/2, 1/3, a) (2),(1,1),(1,1) 2 Has 2-descent
(1/2, 1/3, a) (2,1),(3),(1,1,1) 3 f ∈ C(x)
(1/2, 1/3, a) (22),(3,1),(1,1,2) 4 f ∈ C(x)
(1/2, 1/3, a) (23),(32),(1,1,22) 6 f ∈ C(x)(23),(32),(13, 3) f ∈ C(x)
(1/2, 1/4, a) (2),(2),(1,1) 2 Has 2-descent
(1/2, 1/4, a) (2,2),(4),(14) 4 Has 2-descent
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 20/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Near Belyi Maps
Non parametric Case for near Belyi Maps
Exponent-Difference Ramification type Deg ftype (Pattern)
(1/2, 1/3, 1/7) (25),(33,1),(7,13) 10 f ∈ C(x)(1/2, 1/3, 2/7) (25),(33,1),(7,13) 10 f ∈ C(x)(1/2, 1/3, 3/7) (25),(33,1),(7,13) 10 f ∈ C(x)
(1/2, 1/3, 1/7) (26),(34),(7,13,2) 12 f ∈ C(x)(1/2, 1/3, 2/7) (26),(34),(7,13,2) 12 f ∈ C(x)(1/2, 1/3, 3/7) (26),(34),(7,13,2) 12 f ∈ C(x)
(1/2, 1/3, 1/7) (29),(36),(7,32,22) 18 Has 2-descent(1/2, 1/3, 2/7) (29),(36),(7,32,22) 18 Has 2-descent(1/2, 1/3, 3/7) (29),(36),(7,32,22) 18 Has 2-descent
(1/2, 1/3, 1/8) (26),(34),(8,14) 12 f ∈ C(x)
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 21/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
Example 3
L = (x − 37)∂2 + ∂ + 9(9+x)16(x2+3)
L has the ramification type (2,1),(3),(1,1,1). By searching for thetable, we know this corresponds to one of the Parametric case.In this case f has degree 3, and the exponent-difference type is( 1
2 ,13 , 0).
We calculate f and get:
f =8(9x + 10)2
(3x − 13)3
Furthermore, we find the 2F1-type solution of L as follows:
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 22/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
Example 3
L = (x − 37)∂2 + ∂ + 9(9+x)16(x2+3)
L has the ramification type (2,1),(3),(1,1,1). By searching for thetable, we know this corresponds to one of the Parametric case.In this case f has degree 3, and the exponent-difference type is( 1
2 ,13 , 0).
We calculate f and get:
f =8(9x + 10)2
(3x − 13)3
Furthermore, we find the 2F1-type solution of L as follows:
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 22/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
Example 3
L = (x − 37)∂2 + ∂ + 9(9+x)16(x2+3)
L has the ramification type (2,1),(3),(1,1,1). By searching for thetable, we know this corresponds to one of the Parametric case.In this case f has degree 3, and the exponent-difference type is( 1
2 ,13 , 0).
We calculate f and get:
f =8(9x + 10)2
(3x − 13)3
Furthermore, we find the 2F1-type solution of L as follows:
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 22/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
Example 3, Continued...
y1 =(x2 + 3)
2(3x − 13)54
· 2F1
(1/12, 5/12
1/28(9x+10)2
(3x−13)3
)− 5(9x + 10)(x − 37)(x2 + 3)
(3x − 13)174
· 2F1
(13/12, 17/12
3/28(9x+10)2
(3x−13)3
)y2 = · · ·
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 23/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
Barkatou, M. A., and Pflugel, E.On the Equivalence Problem of Linear Differential Systemsand its Application for Factoring Completely ReducibleSystems. In ISSAC 1998, 268–275.
Bronstein, M.An improved algorithm for factoring linear ordinary differentialoperators. In ISSAC 1994, 336–340.
Compoint, E.,van der Put, M.,and Weil, J.A.Effective descent for differential operators. J. Algebra.324(2010), 146–158.
Debeerst, R, van Hoeij, M, and Koepf. W.Solving Differential Equations in Terms of Bessel Functions. InISSAC 2008, 39–46.
Fang, T.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 23/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
Implementation and examples for 2-descent.www.math.fsu.edu/∼tfang/2descentprogram/
Fang, T. van Hoeij, M.2-descent for second order linear differential equations. InISSAC 2011, 107–114.
van Hoeij, M., and Vidunas, R.All non-Liouvillian 2F1-solvable Heun equations with pullbacksin C(x). www.math.fsu.edu/∼hoeij/files/Heun/TextFormat/
van Hoeij, M.Factorization of Linear Differential Operators. PhD thesis,Universiteit Nijmegen, 1996.
van Hoeij, M.Implementation for finding equivalence map.www.math.fsu.edu/∼hoeij/files/equiv.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 23/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
van Hoeij, M.Solving Third Order Linear Differential Equations in Terms ofSecond Order Equations. In ISSAC 2007, 355–360.Implementation at:www.math.fsu.edu/∼hoeij/files/ReduceOrder
van Hoeij, M, and Cremona, J.Solving conics over function fields. J. de Theories des Nombresde Bordeaux.18(2006), 595–606.
van Hoeij, M, and van der Put, M.Descent for differential modules and skew fields. J. Algebra.296(2006), 18–55.
van der Hoeven, J.Around the Numeric-Symbolic Computation of DifferentialGalois Groups. J. Symb. Comp. 42 (2007), 236–264.
van der Put, M., and Singer, M. F.Tingting Fang Solving Differential Equations January 6th, 2012 Slide 23/ 23
Introduction 2-descent C2× C2-descent Tables Methods
Example
Galois Theory of Linear Differential Equations, vol. 328 of ASeries of Comprehensive Studies in Mathematics. Springer,Berlin, 2003.
Tingting Fang Solving Differential Equations January 6th, 2012 Slide 23/ 23