RationalNormalForms
PolynomialNormalForm
construct the polynomial normal form of a rational function
Calling Sequence
Parameters
Description
Examples
References
PolynomialNormalForm(F, x)
F
-
rational function in x
x
variable
The PolynomialNormalForm(F,x) function constructs the polynomial normal form for F, where F is a rational function in x over a field of characteristic 0.
A sequence of four elements z,a,b,c, where z is an element in K and a,b,c are monic polynomials over K such that the following three conditions are satisfied, is returned: F=z⁢a⁢E⁡cb⁢c. gcd⁡a,Ek⁡b=1⁢for all⁢non−negative integers⁢k. gcd⁡a,c=1,gcd⁡b,E⁡c=1.
Note: E is the automorphism of K(x) defined by {E(f(x)) = f(x+1)}.
This function is part of the RationalNormalForms package, and so it can be used in the form PolynomialNormalForm(..) only after executing the command with(RationalNormalForms). However, it can always be accessed through the long form of the command by using RationalNormalForms[PolynomialNormalForm](..).
with⁡RationalNormalForms:
F≔32⁢n⁢n+2⁢3⁢n+2⁢3⁢n+4n−1⁢2⁢n+9⁢n+42
F≔3⁢n⁢n+2⁢3⁢n+2⁢3⁢n+42⁢n−1⁢2⁢n+9⁢n+42
z,a,b,c≔PolynomialNormalForm⁡F,n
z,a,b,c≔274,n+2⁢n+23⁢n+43,n+92⁢n+42,n−1
Check the results.
Condition 1:
evalb⁡F=normal⁡z⁢ab⁢subs⁡n=n+1,cc
true
Condition 2:
LREtoolsdispersion⁡b,a,n
FAIL
Condition 3:
gcd⁡a,c,gcd⁡b,subs⁡n=n+1,c
1,1
Petkovsek, M.; Wilf, H.; and Zeilberger, D. A=B. Wellesley, Massachusetts: A. K. Peters Ltd., 1996.
See Also
evalb
gcd
LREtools
RationalNormalForms[RationalCanonicalForm]
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