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 $f=\genfrac{}{}{0ex}{}{g}{\phantom{x=h}}|\genfrac{}{}{0ex}{}{\phantom{g}}{x=h}$ 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.

$\mathrm{with}\left(\mathrm{PolynomialTools}\right)\:$

$f≔\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{eval}\left(x^2+2x-1\,x=x^3-x\right)\,x=x^2+3x\right)\right)$

$f≔x^{12}+18x^{11}+135x^{10}+540x^9+1213x^8+1434x^7+623x^6-198x^5-107x^4+60x^3+7x^2-6x-1$

$\mathrm{FunctionalDecomposition}\left(f\,x\right)$

${\left({\left(x^2+3x\right)}^3-x^2-3x\right)}^2+2{\left(x^2+3x\right)}^3-2x^2-6x-1$

$\mathrm{f1}≔\mathrm{expand}\left(\mathrm{eval}\left(x^2+x\exp \left(a\right)-1\,x=x^2-x\right)\right)$

$\mathrm{f1}≔x^4-2x^3+x^2+{\ⅇ}^a{x}^2-x{\ⅇ}^a-1$

$\mathrm{FunctionalDecomposition}\left(\mathrm{f1}\,x\right)$

${\left(x^2-x\right)}^2+\left(x^2-x\right){\ⅇ}^a-1$

$\mathrm{f2}≔\mathrm{expand}\left(\mathrm{eval}\left(-x^3-2x^2-x+1\,x=-y^2+x-1\right)\right)$

$\mathrm{f2}≔y^6-3x{y}^4+3x^2{y}^2+y^4-x^3-2x{y}^2+x^2+1$

$\mathrm{FunctionalDecomposition}\left(\mathrm{f2}\right)$

$-{\left(-y^2+x-1\right)}^3-2{\left(-y^2+x-1\right)}^2+y^2-x+2$

$\mathrm{FunctionalDecomposition}\left(\mathrm{f2}\,'\mathrm{inert}'\right)$

$-{\left(-y^2+x-1\right)}^3-2{\left(-y^2+x-1\right)}^2-\left(-y^2+x-1\right)+1$

The PolynomialTools[FunctionalDecomposition] command was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.