return isomorphic images of Ore polynomials under the Hilbert twist reduction - Maple Help

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OreTools[MathOperations][HilbertTwistReduction] - return isomorphic images of Ore polynomials under the Hilbert twist reduction

OreTools[MathOperations][InverseOfHilbertTwistReduction] - return pre-images of Ore polynomials under the HilbertTwistReduction

OreTools[MathOperations][AccurateIntegration] - check for the existence of a primitive element, and perform accurate integration

Calling Sequence

HilbertTwistReduction(P,  A, 'B')

InverseOfHilbertTwistReduction(P, A)

AccurateIntegration(L, A)

Note: An Ore polynomial ring B is of Hilbert's twist type if its (pseudo) derivation maps everything to zero. For an Ore polynomial ring A with nontrivial automorphism, there is a ring isomorphism from A onto the ring B of Hilbert's twist type whose automorphism is the same as the A's. The isomorphism is called the Hilbert twist reduction.

Parameters

P

-

Ore polynomial or a list of Ore polynomials; to define an Ore polynomial, use the OrePoly structure.

L

-

Ore polynomial.

A

-

Ore ring with nontrivial automorphism; to define an Ore algebra, use the SetOreRing function.

B

-

(optional) unevaluated name.

Description

• 

The HilbertTwistReduction(P, A, B) calling sequence returns the image of P under the Hilbert twist reduction.  If the (optional) third argument B is present, it is assigned to the Ore ring whose automorphism is the same as the A's and whose (pseudo) derivation sends everything to zero.

• 

The InverseOfHilbertTwistReduction(P, A) calling sequence returns the pre-image of P under the Hilbert twist reduction. Note that A is the source ring of the Hilbert twist reduction.

• 

Let A be the shift, q-shift, or differential algebra. The AccurateIntegration(L, A) calling sequence performs accurate integration, which solves the following problem: Let y satisfy L(y)=0 and g satisfy lambda(g)=y, where lambda means the usual derivative in the differential case, the difference operator in the shift case, and the q-difference operator in the q-shift case. The function builds an annihilator S (represented as an OrePoly structure) for g of the same degree as that of L, and an operator K such that g=K(y) if both exist. Otherwise, it returns Lλ.

Examples

withOreTools:

withOreTools[MathOperations]:

withOreTools[Properties]:

Define an Ore ring.

A := SetOreRing(n, 'difference',
  'sigma' = proc(p, x) eval(p, x=x+1) end,
  'sigma_inverse' = proc(p, x) eval(p, x=x-1) end,
  'delta' = proc(p, x) eval(p, x=x+1) - p end,
  'theta1' = 0);

A:=UnivariateOreRingn,difference

(1)

P1:=OrePolyn,n1

P1:=OrePolyn,n1

(2)

Q1:=HilbertTwistReductionP1,A,'B'

Q1:=OrePoly1,n1

(3)

v:=GetVariableB

v:=n

(4)

GetSigmaBsv,v

sn+1

(5)

GetdeltaBsv

0

(6)

R1:=InverseOfHilbertTwistReductionQ1,A

R1:=OrePolyn,n1

(7)

Examples of AccurateIntegration:

A:=SetOreRingx,'differential'

A:=UnivariateOreRingx,differential

(8)

L:=OrePoly23x+1x24+27x,2x4+27x,1

L:=OrePoly23x+1x24+27x,2x4+27x,1

(9)

AccurateIntegrationL,A

OrePoly1,1180162x29x2x,11803x14+27x,OrePoly1180162x29x2x,11803x14+27x

(10)

A:=SetOreRingn,'shift'

A:=UnivariateOreRingn,shift

(11)

L:=OrePoly1,2,2,1

L:=OrePoly1,2,2,1

(12)

AccurateIntegrationL,A

OrePoly12,1,1,12,OrePoly32,12,12

(13)

See Also

OreTools, OreTools/OreAlgebra, OreTools/OrePoly, OreTools[MathOperations], OreTools[Properties], OreTools[SetOreRing]

References

  

Abramov, S.A., and van Hoeij, M. "Integration of Solutions of Linear Functional Equations." Integral Transformations and Special Functions. Vol. 8 No. 1-2. (1999): 3-12.


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