MultivariatePowerSeries/EvaluateAtOrigin - Maple Help

MultivariatePowerSeries

 EvaluateAtOrigin
 Evaluate the coefficients of a univariate polynomial over power series at the origin

 Calling Sequence EvaluateAtOrigin(u)

Parameters

 u - univariate polynomial over power series generated by this package

Description

 • The  command EvaluateAtOrigin(u) returns the image of the univariate polynomial u modulo  the maximal ideal of the ring of power series of the coefficients of u. In other words, the command EvaluateAtOrigin(u)  returns a univariate polynomial (over the complex numbers) obtained from u by specializing to zero every variable in the coefficients of u.
 • 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

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

We define a univariate polynomial over power series and evaluate it at the origin (in this case, at $x=0$).

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left({z}^{2}+z\right),z\right):$
 > $\mathrm{EvaluateAtOrigin}\left(f\right)$
 ${{z}}^{{3}}{-}{6}{}{{z}}^{{2}}{+}{11}{}{z}{-}{6}$ (1)

We define another univariate polynomial over power series and evaluate it at the origin (in this case, at $x=0$ and $y=0$).

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

Compatibility

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