MakeMonic - Maple Help

Algebraic

 MakeMonic
 rewrite a RootOf in terms of a monic RootOf

 Calling Sequence MakeMonic(r)

Parameters

 r - algebraic expression

Description

 • MakeMonic expresses a RootOf in terms of a monic RootOf. It pulls the leading coefficient of the defining polynomial out of the RootOf and into the denominator.
 • MakeMonic only works on indexed, labeled, or one-argument RootOfs. For RootOfs with a numerical approximation or a range as selector, MakeMonic returns the input unchanged.
 • Nested RootOfs that are indexed, labeled, or one-argument, are handled recursively.
 • If r is not a RootOf or not of type algext, it is returned unchanged.
 • For indexed RootOfs, the leading coefficient is pulled out only if its signum is $1$ or $-1$.

Examples

 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(\mathrm{RootOf}\left(y{x}^{3}-1,x\right)\right)$
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{-}{{y}}^{{2}}\right)}{{y}}$ (1)
 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(1+\mathrm{RootOf}\left(y{x}^{3}-1,x\right)\right)$
 ${1}{+}{\mathrm{RootOf}}{}\left({y}{}{{\mathrm{_Z}}}^{{3}}{-}{1}\right)$ (2)
 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(\mathrm{RootOf}\left(2\mathrm{sin}\left(x\right)-1\right)\right)$
 ${\mathrm{RootOf}}{}\left({2}{}{\mathrm{sin}}{}\left({\mathrm{_Z}}\right){-}{1}\right)$ (3)
 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(\mathrm{RootOf}\left(y{x}^{3}-1,x,\mathrm{index}=1\right)\right)$
 ${\mathrm{RootOf}}{}\left({y}{}{{\mathrm{_Z}}}^{{3}}{-}{1}{,}{\mathrm{index}}{=}{1}\right)$ (4)
 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(\mathrm{RootOf}\left(y{x}^{3}-1,x,\mathrm{index}=1\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{-}{{y}}^{{2}}{,}{\mathrm{index}}{=}{1}\right)}{{y}}$ (5)
 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(\mathrm{RootOf}\left(y{x}^{3}-1,x,\mathrm{index}=1\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}y<0$
 ${-}\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{+}{{y}}^{{2}}{,}{\mathrm{index}}{=}{1}\right)}{{y}}$ (6)
 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(\mathrm{RootOf}\left(5{y}^{2}-\mathrm{RootOf}\left(3{x}^{2}+1,\mathrm{index}=1\right),\mathrm{index}=1\right)\right)$
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{15}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{3}{,}{\mathrm{index}}{=}{1}\right){,}{\mathrm{index}}{=}{1}\right)}{{15}}$ (7)

Note that RootOf itself tries to get rid of non-integral leading coefficients by inverting them:

 > $\mathrm{RootOf}\left(\mathrm{RootOf}\left({\mathrm{_Z}}^{2}-2\right){\mathrm{_Z}}^{2}-2\right)$
 ${\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right)\right)$ (8)

In the next example, the leading coefficient is not invertible, and after pulling it out there is a zero divisor in the denominator:

 > $f≔\mathrm{RootOf}\left(\mathrm{RootOf}\left({x}^{2}-x\right){y}^{2}-2\right)$
 ${f}{≔}{\mathrm{RootOf}}{}\left({\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{_Z}}\right){}{{\mathrm{_Z}}}^{{2}}{-}{2}\right)$ (9)
 > $\mathrm{Algebraic}:-\mathrm{MakeMonic}\left(f\right)$
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{_Z}}\right)\right)}{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{_Z}}\right)}$ (10)