compoly - Maple Programming Help

compoly

determine a possible composition of a polynomial

 Calling Sequence compoly(r) compoly(r, x) compoly(r, {x, ... })

Parameters

 r - polynomial x - variable on which the composition will be made

Description

 • The function compoly returns a pair $p\left(x\right)$, $x=q\left(x\right)$ such that subs(x=q(x), p(x)) is equal to r, the input polynomial. If such a pair cannot be found, it returns FAIL. $p\left(x\right)$ and $q\left(x\right)$ are nonlinear polynomials and $q\left(x\right)$ has a low-degree in x greater or equal to 1.
 • When compoly is called without additional arguments, it is equivalent to being called with a second argument which is indets(r). When the second argument is a set with more than one variable, then a multivariate composition is attempted. In this case, the result has the same form, but the second polynomial is multivariate, that is, $p\left(x\right)$, $x=q\left(x,...\right)$.  For the multivariate case, the second polynomial, $q\left(x\right)$, may be of degree 1.
 • Note that the composition may not be unique. In particular, if $p\left(x\right)$, $x=q\left(x,...\right)$ is a composition, then we can find another $p\left(x\right)$ such that replacing $q\left(x,...\right)$ by  $cq\left(x,\dots \right)+b$ will also result in a valid composition. This non-determinacy is eliminated by selecting c and b such that the q polynomial has integer content 1 and its independent term is 0.

Examples

 > $\mathrm{compoly}\left({x}^{6}-9{x}^{5}+27{x}^{4}-27{x}^{3}-2{x}^{2}y+6xy+1,x\right)$
 ${{x}}^{{3}}{-}{2}{}{x}{}{y}{+}{1}{,}{x}{=}{{x}}^{{2}}{-}{3}{}{x}$ (1)
 > $\mathrm{compoly}\left({x}^{4}-3{x}^{3}-x+5,x\right)$
 ${\mathrm{FAIL}}$ (2)
 > $\mathrm{compoly}\left({x}^{2}+2xy-7x+{y}^{2}-7y+16\right)$
 ${{x}}^{{2}}{-}{7}{}{x}{+}{16}{,}{x}{=}{x}{+}{y}$ (3)
 > $\mathrm{compoly}\left({x}^{4}+4{x}^{3}{y}^{3}+6{x}^{2}{y}^{6}+4x{y}^{9}+{y}^{12}+x+{y}^{3}-1,\left\{x,y\right\}\right)$
 ${{x}}^{{4}}{+}{x}{-}{1}{,}{x}{=}{{y}}^{{3}}{+}{x}$ (4)