PolynomialNormalForm - Maple Help

RationalNormalForms

 PolynomialNormalForm
 construct the polynomial normal form of a rational function

 Calling Sequence PolynomialNormalForm(F, x)

Parameters

 F - rational function in x x - variable

Description

 • 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: $F=\frac{zaE\left(c\right)}{bc}.$  $\mathrm{gcd}\left(a,{E}^{k\left(b\right)}\right)=1\mathrm{for all}\mathrm{non}-\mathrm{negative integers}k.$ $\mathrm{gcd}\left(a,c\right)=1,\mathrm{gcd}\left(b,E\left(c\right)\right)=1.$
 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RationalNormalForms}\right):$
 > $F≔\frac{\frac{3}{2}n\left(n+2\right)\left(3n+2\right)\left(3n+4\right)}{\left(n-1\right)\left(2n+9\right){\left(n+4\right)}^{2}}$
 ${F}{≔}\frac{{3}{}{n}{}\left({n}{+}{2}\right){}\left({3}{}{n}{+}{2}\right){}\left({3}{}{n}{+}{4}\right)}{{2}{}\left({n}{-}{1}\right){}\left({2}{}{n}{+}{9}\right){}{\left({n}{+}{4}\right)}^{{2}}}$ (1)
 > $z,a,b,c≔\mathrm{PolynomialNormalForm}\left(F,n\right)$
 ${z}{,}{a}{,}{b}{,}{c}{≔}\frac{{27}}{{4}}{,}\left({n}{+}{2}\right){}\left({n}{+}\frac{{2}}{{3}}\right){}\left({n}{+}\frac{{4}}{{3}}\right){,}\left({n}{+}\frac{{9}}{{2}}\right){}{\left({n}{+}{4}\right)}^{{2}}{,}{n}{-}{1}$ (2)

Check the results.

Condition 1:

 > $\mathrm{evalb}\left(F=\mathrm{normal}\left(\frac{z\left(\frac{a}{b}\right)\mathrm{subs}\left(n=n+1,c\right)}{c}\right)\right)$
 ${\mathrm{true}}$ (3)

Condition 2:

 > $\mathrm{LREtools}\left[\mathrm{dispersion}\right]\left(b,a,n\right)$
 ${\mathrm{FAIL}}$ (4)

Condition 3:

 > $\mathrm{gcd}\left(a,c\right),\mathrm{gcd}\left(b,\mathrm{subs}\left(n=n+1,c\right)\right)$
 ${1}{,}{1}$ (5)

References

 Petkovsek, M.; Wilf, H.; and Zeilberger, D. A=B. Wellesley, Massachusetts: A. K. Peters Ltd., 1996.