FunctionalDecomposition - Maple Help
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PolynomialTools

 FunctionalDecomposition
 compute a functional decomposition of a polynomial

 Calling Sequence FunctionalDecomposition(f) FunctionalDecomposition(f, v) FunctionalDecomposition(f, v, inert)

Parameters

 f - multivariate polynomial v - name or list or set of names

Description

 • This function computes a functional decomposition of the polynomial f. That is, it computes g(x) and h and rewrites the output as a composition $f=\genfrac{}{}{0}{}{g}{\phantom{x=h}}|\genfrac{}{}{0}{}{\phantom{g}}{x=h}$ and the process is repeated on g and h until they can not be functionally decomposed further. This decomposition is not unique.
 • This function currently just calls compoly repeatedly and constructs the decomposition as a single unexpanded polynomial. If no decomposition is found, f is returned unaltered.
 • If f is not of type polynom then frontend is used before performing polynomial calculations.
 • The inert option adds () calls around $x$ so that the linear part of the polynomial and pure monomial substitutions will not be expanded by automatic simplification. The output will look like a polynomial, but will not be true polynomial unless expand is called to remove the () similar to the output of ifactor.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialTools}\right):$
 > $f≔\mathrm{expand}\left(\genfrac{}{}{0}{}{\genfrac{}{}{0}{}{\left({x}^{2}+2x-1\right)}{\phantom{x={x}^{3}-x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\left({x}^{2}+2x-1\right)}}{x={x}^{3}-x}}{\phantom{x={x}^{2}+3x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\genfrac{}{}{0}{}{\left({x}^{2}+2x-1\right)}{\phantom{x={x}^{3}-x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\left({x}^{2}+2x-1\right)}}{x={x}^{3}-x}}}{x={x}^{2}+3x}\right)$
 ${f}{≔}{{x}}^{{12}}{+}{18}{}{{x}}^{{11}}{+}{135}{}{{x}}^{{10}}{+}{540}{}{{x}}^{{9}}{+}{1213}{}{{x}}^{{8}}{+}{1434}{}{{x}}^{{7}}{+}{623}{}{{x}}^{{6}}{-}{198}{}{{x}}^{{5}}{-}{107}{}{{x}}^{{4}}{+}{60}{}{{x}}^{{3}}{+}{7}{}{{x}}^{{2}}{-}{6}{}{x}{-}{1}$ (1)
 > $\mathrm{FunctionalDecomposition}\left(f,x\right)$
 ${\left({\left({{x}}^{{2}}{+}{3}{}{x}\right)}^{{3}}{-}{{x}}^{{2}}{-}{3}{}{x}\right)}^{{2}}{+}{2}{}{\left({{x}}^{{2}}{+}{3}{}{x}\right)}^{{3}}{-}{2}{}{{x}}^{{2}}{-}{6}{}{x}{-}{1}$ (2)
 > $\mathrm{f1}≔\mathrm{expand}\left(\genfrac{}{}{0}{}{\left({x}^{2}+x{ⅇ}^{a}-1\right)}{\phantom{x={x}^{2}-x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\left({x}^{2}+x{ⅇ}^{a}-1\right)}}{x={x}^{2}-x}\right)$
 ${\mathrm{f1}}{≔}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{{ⅇ}}^{{a}}{}{{x}}^{{2}}{-}{x}{}{{ⅇ}}^{{a}}{-}{1}$ (3)
 > $\mathrm{FunctionalDecomposition}\left(\mathrm{f1},x\right)$
 ${\left({{x}}^{{2}}{-}{x}\right)}^{{2}}{+}\left({{x}}^{{2}}{-}{x}\right){}{{ⅇ}}^{{a}}{-}{1}$ (4)
 > $\mathrm{f2}≔\mathrm{expand}\left(\genfrac{}{}{0}{}{\left(-{x}^{3}-2{x}^{2}-x+1\right)}{\phantom{x=-{y}^{2}+x-1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\left(-{x}^{3}-2{x}^{2}-x+1\right)}}{x=-{y}^{2}+x-1}\right)$
 ${\mathrm{f2}}{≔}{{y}}^{{6}}{-}{3}{}{x}{}{{y}}^{{4}}{+}{3}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{-}{{x}}^{{3}}{-}{2}{}{x}{}{{y}}^{{2}}{+}{{x}}^{{2}}{+}{1}$ (5)
 > $\mathrm{FunctionalDecomposition}\left(\mathrm{f2}\right)$
 ${-}{\left({-}{{y}}^{{2}}{+}{x}{-}{1}\right)}^{{3}}{-}{2}{}{\left({-}{{y}}^{{2}}{+}{x}{-}{1}\right)}^{{2}}{+}{{y}}^{{2}}{-}{x}{+}{2}$ (6)
 > $\mathrm{FunctionalDecomposition}\left(\mathrm{f2},'\mathrm{inert}'\right)$
 ${-}{\left({-}{{y}}^{{2}}{+}{x}{-}{1}\right)}^{{3}}{-}{2}{}{\left({-}{{y}}^{{2}}{+}{x}{-}{1}\right)}^{{2}}{-}\left({-}{{y}}^{{2}}{+}{x}{-}{1}\right){+}{1}$ (7)

Compatibility

 • The PolynomialTools[FunctionalDecomposition] command was introduced in Maple 2022.
 • For more information on Maple 2022 changes, see Updates in Maple 2022.