closure of a q-shift operator with polynomial coefficients - Maple Help

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QDifferenceEquations[Closure] - closure of a q-shift operator with polynomial coefficients

Calling Sequence

Closure(L, Qx, x, q, p, func, options)

Parameters

L

-

polynomial in Qx with coefficients which are polynomials in x over the field of rational functions in q

Qx

-

name, variable denoting the q-shift operator xqx 

x

-

variable name

q

-

either a variable name, or a nonzero constant, or an equation of the form name=constant

p

-

(optional) set of irreducible polynomials in x, or a single such polynomial

func

-

(optional) procedure

options

-

(optional) equation(s) of the form 'keyword'=value, where the keyword is either order or maximal

Returns

• 

list of polynomials in Qx with coefficients which are polynomials in x over the field of rational functions in q

Description

• 

Let k be a field of characteristic 0. Denote by F the q-shift polynomial ring consisting of elements, each of which is a polynomial in Qx, with coefficients which are polynomials in x over kq. For a given operator LF, the Closure(L,Qx,x,q) calling sequence constructs a closure of L in the q-shift polynomial ring F.

  

The output is a list of elements in F. Each element R in this list represents a generator of the closure of L. For example, there exists fkqx and PF, such that the torsion relation P.L=f.R holds.

• 

The Closure(L,Qx,x q,p) calling sequence constructs a local closure of L at the irreducible(s) p. The output is a list of generators of the local closure of L.  For example, each element R in the list is such that the torsion relation P.L=gi.R holds for some operator P, where gp and i.

• 

The parameter q does not have to be a variable. A constant value, such as q=2 is possible as well, including the case of a root of unity.

• 

The optional argument func, if specified, is applied to the coefficients of the result with respect to Qx; typical examples are expand or factor.

• 

Note that setting infolevel[Closure]:=3 will cause some diagnostics to be printed during the computation.

Options

• 

'order'=o, where o is a monomial order

  

If this option is given, the Groebner[Basis] command, with respect to the given monomial order, will be applied to the computed closure.

• 

'maximal'=truefalse (default: false)

  

This option only has an effect if a local closure is requested. If maximal=true (or maximal for short) is specified, then each element R in the output list is such that the torsion relation P.L=gqjxi.R holds for some operator P, where gp, i, and j. In other words, all generators for the q-shift-equivalence class(es) represented by p are computed and returned.

Examples

withQDifferenceEquations

AccurateQSummation,AreSameSolution,Closure,Desingularize,ExtendSeries,IsQHypergeometricTerm,IsSolution,PolynomialSolution,QBinomial,QBrackets,QDispersion,QECreate,QEfficientRepresentation,QFactorial,QGAMMA,QHypergeometricSolution,QMultiplicativeDecomposition,QPochhammer,QPolynomialNormalForm,QRationalCanonicalForm,QSimpComb,QSimplify,RationalSolution,RegularQPochhammerForm,SeriesSolution,UniversalDenominator,Zeilberger

(1)

Compute a closure of the following linear q-shift operator when q=12:

L:=x3qx3Qx+q2x32q3x3

L:=x3qx3Qx+q2x32q3x3

(2)

C:=ClosureL,Qx,x,q=12

C:=12x292x+9Qx+1128x338x2+458x27,34x+9Qx3+31024x29128x40516Qx2+92048x2+81256x8116Qx,863x12821Qx4+14032x2556x+1917Qx3+399128x634Qx2+512048x2+243128x1354Qx,32x+9Qx2+3256x2964x40516Qx9512x2+81128x8116,1663x12821Qx3+11008x2528x+1917Qx2+39964x634Qx51512x2+24364x1354

(3)

Compute a local closure of L, now with a symbolic q, at p=13q3x+1, with factored coefficients:

p:=1q3x3+1

p:=13q3x+1

(4)

C:=ClosureL,Qx,x,q,p,factor

C:=x3qx3Qx+q2x32q3x3,3q2x+9Qx3+3q10x29q7x+18q6x+9q5x+9q4x27q49q3x+27q327Qx2+9q1q3x3q4x3q3Qx,q4x3Qx4q+12q2+q+1q12q+q13x2+3q10x6q8x3q7x+9q7+9q69q518q49q3+9q2+9q+9Qx3q+12q2+q+1q12q+9q2+q+1q6xq4x2q3x+6q3+3q23Qx2q+19q4x33q5x+3q4x+2q3x6q29q9q3Qxq+1

(5)

Compute a local closure of L at the q-shift equivalence class represented by p:

C:=ClosureL,Qx,x,q,p,'maximal',factor

C:=x3qx3Qx+q2x32q3x3,3q2x+9Qx3+3q10x29q7x+18q6x+9q5x+9q4x27q49q3x+27q327Qx2+9q1q3x3q4x3q3Qx,q4x3Qx4q+12q2+q+1q12q+q13x2+3q10x6q8x3q7x+9q7+9q69q518q49q3+9q2+9q+9Qx3q+12q2+q+1q12q+9q2+q+1q6xq4x2q3x+6q3+3q23Qx2q+19q4x33q5x+3q4x+2q3x6q29q9q3Qxq+1,3qx+9Qx2+3q8x29q6x+18q5x+9q4x27q4+9q3x+27q39q2x27Qx+9q1q2x3q3x3q3,q3x3Qx3q+12q2+q+1q12q+q11x2+3q9x6q7x+9q73q6x+9q69q518q49q3+9q2+9q+9Qx2q+12q2+q+1q12q+9q2+q+1q5xq3x+6q32q2x+3q23Qxq+19q3x33q4x+3q3x+2q2x6q29q9q3q+1

(6)

Verify the torsion property:

A:=OreTools:-SetOreRingx,q,'qshift':

L:=OreTools:-Converters:-FromPolyToOrePolyL,Qx

L:=OrePolyq2x32q3x3,x3qx3

(7)

R2:=OreTools:-Converters:-FromPolyToOrePolyC2,Qx

R2:=OrePoly0,9q1q3x3q4x3q3,3q10x29q7x+18q6x+9q5x+9q4x27q49q3x+27q327,3q2x+9

(8)

OreTools:-Remainder'right'R2,L,A,'P2'

OrePoly0

(9)

P2:=OreTools:-Converters:-FromOrePolyToPolyP2,Qx

P2:=9q3q1Qxq3x33Qx2q3x3

(10)

denomP2

q3x3

(11)

R4:=OreTools:-Converters:-FromPolyToOrePolyC4,Qx

R4:=OrePoly9q1q2x3q3x3q3,3q8x29q6x+18q5x+9q4x27q4+9q3x+27q39q2x27,3qx+9

(12)

OreTools:-Remainder'right'R4,L,A,'P4'

OrePoly0

(13)

P4:=OreTools:-Converters:-FromOrePolyToPolyP4,Qx

P4:=9q3q1q2x33Qxq2x3

(14)

denomP4

q2x3

(15)

This is, in fact, a negative q-shift of p:

numerpx=xq|px=xq

q2x+3

(16)

Compute a closure of the following operator when q=1, a second root of unity:

L:=x+22+x+1x22Qx

L:=x+22+x+1x22Qx

(17)

infolevel[Closure]:=3:

C:=ClosureL,Qx,x,1

Closure:   "-1 is a 2 root of unity"
Closure:   "compute the matrix representation of the input operator"
Closure:   "the matrix representation is Matrix(2, 2, [[(x+2)^2,(x+1)*(x-2)^2],[(-x+1)*(-x-2)^2*eta,(-x+2)^2]])"
Closure:   "compute the candidate primes and bounds for their multiplicities"
Closure:   "the candidate primes and bounds for their multiplicities are [[x-2, 2], [x+2, 2]]"
Closure:   "compute the local closures"
Closure:   "compute P such that P.L = (x-2)^j.R, 1<=j<=2"
Closure:   "compute P such that P.L = (x+2)^j.R, 1<=j<=2"

C:=x&plus;22&plus;x&plus;1x22Qx&comma;x2x2Qx3&plus;x24x4Qx2&plus;x2Qx&comma;x21Qx3&plus;Qx&comma;x2x2Qx3&plus;x24x4Qx2&plus;x2Qx&comma;x2x&plus;2Qx2&plus;x24x&plus;4Qx&plus;x&plus;2&comma;x&plus;1Qx2&plus;x24x&plus;4Qx&plus;x&plus;3&comma;x2x&plus;2Qx2&plus;x24x&plus;4Qx&plus;x&plus;2

(18)

See Also

Groebner, Ore_algebra, QDifferenceEquations[Desingularize]


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