The command IsUnit(p) returns true if the power series p is invertible, and false otherwise. A power series is invertible if and only if its constant coefficient is nonzero.

The command IsUnit(u) returns true if the univariate polynomial over power series u is invertible, and false otherwise. A univariate polynomial over power series is invertible if the constant coefficient with respect to its main variable (which is a power series) is invertible. To actually compute the inverse, you need to convert u to a PowerSeries first.

This command is supported for univariate polynomials over power series, but not for univariate polynomials over Puiseux series.

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.

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

$a≔\mathrm{PowerSeries}\left(2x+y+3xz\right)\:$

$\mathrm{GetCoefficient}\left(a\,1\right)$

$0$

$\mathrm{IsUnit}\left(a\right)$

$\mathrm{false}$

$b≔\mathrm{GeometricSeries}\left(\left[x\,y\,z\right]\right)$

$b≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y-z}\mathrm{:}1+x+y+z+\mathrm{\dots }\right]$

$\mathrm{GetCoefficient}\left(b\,1\right)$

$1$

$\mathrm{IsUnit}\left(b\right)$

$\mathrm{true}$

$u≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(x\right)\,\mathrm{PowerSeries}\left(1+y\right)\,\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\right]\,z\right)$

$u≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(x\right)\mathrm{+}\left(1+y\right)z\mathrm{+}\left(1+x+y+\mathrm{\dots }\right)z^2\right]$

$\mathrm{IsUnit}\left(u\right)$

$\mathrm{false}$

$v≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1+x\right)\,\mathrm{PowerSeries}\left(1+y\right)\,\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\right]\,z\right)$

$v≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1+x\right)\mathrm{+}\left(1+y\right)z\mathrm{+}\left(1+x+y+\mathrm{\dots }\right)z^2\right]$

$\mathrm{IsUnit}\left(v\right)$

$\mathrm{true}$

$\mathrm{v\_as\_power\_series}≔\mathrm{PowerSeries}\left(v\right)$

$\mathrm{v\_as\_power\_series}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}x+1+\left(1+y\right)z+\frac{z^2}{1-x-y}\mathrm{:}1+x+z+\mathrm{\dots }\right]$

$\frac{1}{\mathrm{v\_as\_power\_series}}$

$\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{x+1+\left(1+y\right)z+\frac{z^2}{1-x-y}}\mathrm{:}1+\mathrm{\dots }\right]$

The MultivariatePowerSeries[IsUnit] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.