MultivariatePowerSeries/IsUnit - Maple Help

MultivariatePowerSeries

 IsUnit
 Determine if a power series of univariate polynomial over power series is a a unit

 Calling Sequence IsUnit(p) IsUnit(u)

Parameters

 p - power series generated by this package u - univariate polynomial over power series generated by this package

Description

 • 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.
 • When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series and univariate polynomials over power series. If you do, you may see invalid results.

Examples

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

Define a power series (as a polynomial). Its constant coefficient is zero, so it is not invertible.

 > $a≔\mathrm{PowerSeries}\left(2x+y+3xz\right):$
 > $\mathrm{GetCoefficient}\left(a,1\right)$
 ${0}$ (1)
 > $\mathrm{IsUnit}\left(a\right)$
 ${\mathrm{false}}$ (2)

Define another power series. Its constant coefficient is one, so it is invertible.

 > $b≔\mathrm{GeometricSeries}\left(\left[x,y,z\right]\right)$
 ${b}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}{-}{z}}{:}{1}{+}{x}{+}{y}{+}{z}{+}{\dots }\right]$ (3)
 > $\mathrm{GetCoefficient}\left(b,1\right)$
 ${1}$ (4)
 > $\mathrm{IsUnit}\left(b\right)$
 ${\mathrm{true}}$ (5)

Define a univariate polynomial over power series, $u$. The constant coefficient with respect to its main variable, $z$, is $x$, which is not invertible. Thus, $u$ is not invertible.

 > $u≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(x\right),\mathrm{PowerSeries}\left(1+y\right),\mathrm{GeometricSeries}\left(\left[x,y\right]\right)\right],z\right)$
 ${u}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({x}\right){+}\left({1}{+}{y}\right){}{z}{+}\left({1}{+}{x}{+}{y}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (6)
 > $\mathrm{IsUnit}\left(u\right)$
 ${\mathrm{false}}$ (7)

Define a univariate polynomial over power series, $v$. The constant coefficient with respect to its main variable, $z$, is $x+1$, which is invertible. Thus, $v$ is invertible. Its inverse is not a polynomial but a power series, so in order to invert $v$, we need to convert it to a power series first.

 > $v≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1+x\right),\mathrm{PowerSeries}\left(1+y\right),\mathrm{GeometricSeries}\left(\left[x,y\right]\right)\right],z\right)$
 ${v}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}\right){+}\left({1}{+}{y}\right){}{z}{+}\left({1}{+}{x}{+}{y}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (8)
 > $\mathrm{IsUnit}\left(v\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{v_as_power_series}≔\mathrm{PowerSeries}\left(v\right)$
 ${\mathrm{v_as_power_series}}{≔}\left[{PowⅇrSⅇriⅇs of}{x}{+}{1}{+}\left({1}{+}{y}\right){}{z}{+}\frac{{{z}}^{{2}}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{z}{+}{\dots }\right]$ (10)
 > $\frac{1}{\mathrm{v_as_power_series}}$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{x}{+}{1}{+}\left({1}{+}{y}\right){}{z}{+}\frac{{{z}}^{{2}}}{{1}{-}{x}{-}{y}}}{:}{1}{+}{\dots }\right]$ (11)

Compatibility

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