MainVariable - Maple Help

MultivariatePowerSeries

 MainVariable
 get the polynomial variable of a univariate polynomial over power series or over Puiseux series

 Calling Sequence MainVariable(u)

Parameters

 u - univariate polynomial over power series or over Puiseux series generated by this package

Description

 • MainVariable(u) returns the polynomial variable of the univariate polynomial over power series or over Puiseux series u.
 • If u is considered as an element of $\mathrm{𝕂}⟦{x}_{1},\dots ,{x}_{n}⟧\left[y\right]$, then $y$ is the main variable.
 • When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series, Puiseux series, and univariate polynomials over these series. If you do, you may see invalid results.

Examples

 > $\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right):$

We construct a univariate polynomial over power series in $\mathrm{𝕂}⟦x⟧\left[z\right]$. Its main variable is $z$.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left({z}^{2}+z\right),z\right):$
 > $\mathrm{MainVariable}\left(f\right)$
 ${z}$ (1)

We can view the same expression as an element of $\mathrm{𝕂}⟦z⟧\left[x\right]$. Then its main variable is $x$.

 > $g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left({z}^{2}+z\right),x\right):$
 > $\mathrm{MainVariable}\left(g\right)$
 ${x}$ (2)

We construct a univariate polynomial over power series in $\mathrm{𝕂}⟦y,z⟧\left[x\right]$. Its main variable is $x$.

 > $h≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(z\right),\mathrm{PowerSeries}\left(y\right),\mathrm{GeometricSeries}\left(\left[y,z\right]\right)\right],x\right):$
 > $\mathrm{MainVariable}\left(h\right)$
 ${x}$ (3)

Create a univariate polynomial over power series from a list of Puiseux series.

 > $g≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (4)
 > $\mathrm{MainVariable}\left(g\right)$
 ${z}$ (5)

Compatibility

 • The MultivariatePowerSeries[MainVariable] command was introduced in Maple 2021.
 • The MultivariatePowerSeries[MainVariable] command was updated in Maple 2023.