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 RationalCanonicalForm
 construct two differential rational canonical forms of a rational function

 Calling Sequence RationalCanonicalForm[1](F, x) RationalCanonicalForm[2](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 RationalCanonicalForm[i](F,x) calling sequence constructs the ith differential rational canonical forms for F, $i=\left\{1,2\right\}$.
 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
 1 $F=R+\frac{\frac{{\partial }}{{\partial }x}V}{V}$.
 2 $\mathrm{gcd}\left(\mathrm{denom}\left(R\right),\mathrm{numer}\left(R\right)-i\frac{ⅆ}{ⅆx}\left(\mathrm{denom}\left(R\right)\right)\right)=1\mathrm{for all}\mathrm{integers}i.$
 • 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 $R=\frac{a}{b}$, $V=\frac{c}{d}$.
 • The use of RationalCanonicalForm[1] is for testing similarity of two given hyperexponential functions. For RationalCanonicalForm[2], the polynomials $b,c,d$ are also pairwise relatively prime. RationalCanonicalForm[2] is used in a reduction algorithm for hyperexponential functions.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $F≔\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^{2}}-\frac{9}{{\left(x-1\right)}^{2}}-\frac{9{x}^{2}+12}{{x}^{3}+4x-2}+\frac{1}{{\left({x}^{3}+4x-2\right)}^{2}}$
 ${F}{≔}\frac{{4}}{{x}{-}{2}}{+}\frac{{4}}{{x}{+}{1}}{-}\frac{{3}}{{\left({x}{+}{1}\right)}^{{2}}}{-}\frac{{9}}{{\left({x}{-}{1}\right)}^{{2}}}{-}\frac{{9}{}{{x}}^{{2}}{+}{12}}{{{x}}^{{3}}{+}{4}{}{x}{-}{2}}{+}\frac{{1}}{{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}$ (1)
 > $\mathrm{R1},\mathrm{V1}≔\mathrm{RationalCanonicalForm}[1]\left(F,x\right)$
 ${\mathrm{R1}}{,}{\mathrm{V1}}{≔}\frac{{-}{12}{}{{x}}^{{8}}{-}{12}{}{{x}}^{{7}}{-}{108}{}{{x}}^{{6}}{-}{48}{}{{x}}^{{5}}{-}{239}{}{{x}}^{{4}}{+}{48}{}{{x}}^{{3}}{-}{50}{}{{x}}^{{2}}{+}{144}{}{x}{-}{47}}{{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}{}{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}{,}\frac{{\left({x}{-}{2}\right)}^{{4}}{}{\left({x}{+}{1}\right)}^{{4}}}{{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{3}}}$ (2)
 > $\mathrm{R2},\mathrm{V2}≔\mathrm{RationalCanonicalForm}[2]\left(F,x\right)$
 ${\mathrm{R2}}{,}{\mathrm{V2}}{≔}\frac{{-}{5}{}{{x}}^{{9}}{-}{16}{}{{x}}^{{8}}{-}{14}{}{{x}}^{{7}}{-}{134}{}{{x}}^{{6}}{+}{39}{}{{x}}^{{5}}{-}{331}{}{{x}}^{{4}}{+}{96}{}{{x}}^{{3}}{+}{32}{}{{x}}^{{2}}{+}{16}{}{x}{-}{7}}{{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}{}{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}{,}{\left({x}{-}{2}\right)}^{{4}}$ (3)
 > $\mathrm{a1},\mathrm{b1},\mathrm{c1},\mathrm{d1}≔\mathrm{RationalCanonicalForm}[1]\left(F,x,'\mathrm{polyform}'\right)$
 ${\mathrm{a1}}{,}{\mathrm{b1}}{,}{\mathrm{c1}}{,}{\mathrm{d1}}{≔}{-}{12}{}{{x}}^{{8}}{-}{12}{}{{x}}^{{7}}{-}{108}{}{{x}}^{{6}}{-}{48}{}{{x}}^{{5}}{-}{239}{}{{x}}^{{4}}{+}{48}{}{{x}}^{{3}}{-}{50}{}{{x}}^{{2}}{+}{144}{}{x}{-}{47}{,}{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}{}{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}{,}{\left({x}{-}{2}\right)}^{{4}}{}{\left({x}{+}{1}\right)}^{{4}}{,}{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{3}}$ (4)

References

 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.