determine if a power series or univariate polynomial over power series is a unit
power series generated by this package
univariate polynomial over power series generated by this package
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.
Define a power series (as a polynomial). Its constant coefficient is zero, so it is not invertible.
a ≔ PowerSeries⁡2⁢x+y+3⁢x⁢z:
Define another power series. Its constant coefficient is one, so it is invertible.
b ≔ GeometricSeries⁡x,y,z
b≔PowⅇrSⅇrⅈⅇs of 11−x−y−z : 1+x+y+z+…
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 ≔ UnivariatePolynomialOverPowerSeries⁡PowerSeries⁡x,PowerSeries⁡1+y,GeometricSeries⁡x,y,z
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 ≔ UnivariatePolynomialOverPowerSeries⁡PowerSeries⁡1+x,PowerSeries⁡1+y,GeometricSeries⁡x,y,z
v_as_power_series ≔ PowerSeries⁡v
v_as_power_series≔PowⅇrSⅇrⅈⅇs of x+1+1+y⁢z+z21−x−y : 1+x+z+…
PowⅇrSⅇrⅈⅇs of 1x+1+1+y⁢z+z21−x−y : 1+…
The MultivariatePowerSeries[IsUnit] command was introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
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