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QDifferenceEquations

  

QRationalCanonicalForm

  

construct four q-rational canonical forms of a rational function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

QRationalCanonicalForm[1](F, q, n)

QRationalCanonicalForm[2](F, q, n)

QRationalCanonicalForm[3](F, q, n)

QRationalCanonicalForm[4](F, q, n)

Parameters

F

-

rational function of n

q

-

name used as the parameter q, usually q

n

-

variable

Description

• 

Let F be a rational function of n over a field K of characteristic 0, q is a nonzero element of K which is not a root of unity. The QRationalCanonicalForm[i](F,q,n) command constructs the ith rational canonical form for F, i=1,2,3,4.

  

If QRationalCanonicalForm is called without an index, the first q-rational canonical form is constructed.

• 

The output is a sequence of 5 elements z,r,s,u,v, called qRNFF, where z is an element of K, and r,s,u,v are monic polynomials over K such that:

1. 

F=zrsQuvuv, gcdu,v=1.

2. 

gcdr,Qks for all integers k.

3. 

u00, v00.

4. 

gcdr,Q.Ev=1, gcds,Qu.v=1.

  

Note: Q is the automorphism of K(n) defined by QFn=Fqn.

• 

The five-tuple z,r,s,u,v that satisfies the four conditions is a strict q-rational normal form for F. The rational function zrs and uv are called the kernel and the shell of the qRNFF, respectively.

• 

Let φ=z,r,s,u,v be any qRNF of a rational function F. Then the degrees of the polynomials r and s are unique, and have minimal possible values in the sense that if Fn=pnQGnqnGn where p, q are polynomials in n, and G is a rational function of n, then degreerdegreep and degreesdegreeq.

• 

Additionally, if i=1 then degreev is minimal; if i=2 then degreeu is minimal; if i=3 then degreeu+degreev is minimal, and under this condition, degreev is minimal; if i=4 then degreeu+degreev is minimal, and under this condition, degreeu is minimal.

Examples

withQDifferenceEquations:

νn+q2q11n+1n+q5q3n+q4q2q3n+q21q12n+q21:

den+q5n+q42q11q4n+1n+q21q2n+q21:

Fνde

F:=q2+nn+1q5q3+nq4q2+nnq3+q21nq12+q21q5+nq4+n2nq4+1q2+n1nq2+q21

(1)

z1,r1,s1,u1,v1QRationalCanonicalForm[1]F,q,n

z1,r1,s1,u1,v1:=1q10,q5q3+nq4q2+n,q5+nn+1q4,n+q21q22q3+n2q4+n2q+nq2+nn+q21q11n+q21q10n+q21q9n+q21q8n+q21q7n+q21q6n+q21q5n+q21q4n+q21q3n+q21qq2+n1,1

(2)

z2,r2,s2,u2,v2QRationalCanonicalForm[2]F,q,n

z2,r2,s2,u2,v2:=q18,n+q21q3n+q21q12,q4+nq5+n,q3+nq4+n,q3+nq2q4q2+n2n+1q3n+1q2n+1qn+1n+q21qq2+n1q5q3+n

(3)

z3,r3,s3,u3,v3QRationalCanonicalForm[3]F,q,n

z3,r3,s3,u3,v3:=q4,q5q3+nn+q21q12,q5+nn+1q4,q3+n2q4+n2q+nq2+nn+q21q2,q3+nqq4q2+n

(4)

z4,r4,s4,u4,v4QRationalCanonicalForm[4]F,q,n

z4,r4,s4,u4,v4:=q12,q5q3+nn+q21q12,q4+nq5+n,q3+nq4+nn+q21q2,n+1q3n+1q2n+1qn+1q3+nqq4q2+n

(5)

Check the result from QRationalCanonicalForm[2].

Condition 1 is satisfied.

normalFz2r2s2subsn=qn,u2v2u2v2,gcdexu2,v2,n

0,1

(6)

Condition 2 is satisfied.

QDispersionr2,s2,q,n,QDispersions2,r2,q,n

FAIL,FAIL

(7)

Condition 3 is satisfied.

u2n=0|u2n=00,normalv2n=0|v2n=00

q70,q217q20

(8)

Condition 4 is satisfied.

gcdexr2,u2subsn=qn,v2,n,gcdexs2,subsn=qn,u2v2,n

1,1

(9)

Degrees of the kernel:

degreer1,n,degreer2,n,degreer3,n,degreer4,n

2,2,2,2

(10)

degrees1,n,degrees2,n,degrees3,n,degrees4,n

2,2,2,2

(11)

The degree of v1 is minimal:

degreev1,n,degreev2,n,degreev3,n,degreev4,n

0,11,2,6

(12)

The degree of u2 is minimal:

degreeu1,n,degreeu2,n,degreeu3,n,degreeu4,n

19,2,7,3

(13)

For i=3,4, the degree of the shell is minimal:

degreeu1,n+degreev1,n,degreeu2,n+degreev2,n,degreeu3,n+degreev3,n,degreeu4,n+degreev4,n

19,13,9,9

(14)

References

  

Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Efficient Representations of (q-)Hypergeometric Terms and the Assignment Problem." Submitted.

  

Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Rational Canonical Forms and Efficient Representations of Hypergeometric Terms." Proc. ISSAC'2003, pp. 7-14. 2003.

See Also

QDifferenceEquations[QDispersion]

QDifferenceEquations[QEfficientRepresentation]

QDifferenceEquations[QMultiplicativeDecomposition]

QDifferenceEquations[QPolynomialNormalForm]

 


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