construct two differential rational canonical forms of a rational function
rational function of x
Let F be a rational function of x over a field K of characteristic 0. The RationalCanonicalForm[i](F,x) calling sequence constructs the ith differential rational canonical forms for F, i=1,2.
If the RationalCanonicalForm command is called without an index, the first differential rational canonical form is constructed.
The output is a sequence of 2 elements R,V, called RationalCanonicalForm(F), where R,V are rational functions over K such that
If the third optional argument, which is the name 'polyform', is given, the output is a sequence of 4 elements a,b,c,d, where a,b,c,d are polynomials over K, b,c,d monic such that R=ab, V=cd.
The use of RationalCanonicalForm is for testing similarity of two given hyperexponential functions. For RationalCanonicalForm, the polynomials b,c,d are also pairwise relatively prime. RationalCanonicalForm is used in a reduction algorithm for hyperexponential functions.
F ≔ 4x−2+4x+1−3x+12−9x−12−9⁢x2+12x3+4⁢x−2+1x3+4⁢x−22
R1,V1 ≔ RationalCanonicalForm⁡F,x
R1,V1 ≔ −12⁢x8−12⁢x7−108⁢x6−48⁢x5−239⁢x4+48⁢x3−50⁢x2+144⁢x−47x+12⁢x−12⁢x3+4⁢x−22,x−24⁢x+14x3+4⁢x−23
R2,V2 ≔ RationalCanonicalForm⁡F,x
R2,V2 ≔ −5⁢x9−16⁢x8−14⁢x7−134⁢x6+39⁢x5−331⁢x4+96⁢x3+32⁢x2+16⁢x−7x+12⁢x−12⁢x3+4⁢x−22,x−24
a1,b1,c1,d1 ≔ RationalCanonicalForm⁡F,x,'polyform'
a1,b1,c1,d1 ≔ −12⁢x8−12⁢x7−108⁢x6−48⁢x5−239⁢x4+48⁢x3−50⁢x2+144⁢x−47,x+12⁢x−12⁢x3+4⁢x−22,x−24⁢x+14,x3+4⁢x−23
Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press, (2004): 183-190.
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