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PolynomialNormalForm

  

construct the differential polynomial normal form of a rational function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

PolynomialNormalForm(F, x)

Parameters

F

-

rational function of x

x

-

variable

Description

• 

Let F be a rational function of x over a field K of characteristic 0. The PolynomialNormalForm(F,x) command constructs the differential polynomial normal form for F.

• 

The output is a sequence of 3 elements a,b,c where a,b,c are polynomials over K such that:

1. 

 F=ab+ⅆⅆxcc. 

2. 

gcdb,aixb=1 for all non-negative integers i.

3. 

gcdb,c=1.

Examples

withDEtools:

F4x2+4x+13x+129x129x2+12x3+4x2+1x3+4x22

F:=4x2+4x+13x+129x129x2+12x3+4x2+1x3+4x22

(1)

a,b,cPolynomialNormalFormF,x

a,b,c:=5x916x814x7134x6+39x5331x4+96x3+32x2+16x7,x+12x12x3+4x22,x24

(2)

Check the result:

nFab+xcc:

TestzeronormalFnF

true

(3)

resresultantb,ajxb,x:

Hselecttype,solveres,j,'nonnegint':

evalbH=andgcdb,c=1

true

(4)

References

  

Almkvist, G, and Zeilberger, D. "The method of differentiating under the integral sign." Journal of Symbolic Computation. Vol. 10. (1990): 571-591.

See Also

DEtools[Gosper]

DEtools[RationalCanonicalForm]

SumTools[Hypergeometric][PolynomialNormalForm]

 


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