The calling sequence UnivariatePolynomialOverPowerSeries(lp, v) creates a univariate polynomial over power series with main variable v and with coefficients that are power series or Puiseux series from lp. The degree of the resulting polynomial is equal to the length of lp minus one. The coefficient of v^(i-1) is the i-th element of lp. In particular, the first element of lp is the constant coefficient. The main variable, v, cannot occur in any of the power series in lp.

The calling sequences UnivariatePolynomialOverPowerSeries(p, v) and UnivariatePolynomialOverPowerSeries(r, v) create univariate polynomials with power series or Puiseux series coefficients. The former returns the same result as UnivariatePolynomialOverPowerSeries(lp, v) where lp := [seq(PowerSeries(coeff(p,v,i)),i=0..degree(p,v))] or lp := [seq(PowerSeries(coeff(p,v,i)),i=0..degree(p,v))]. For the latter, r is a rational function where v only occurs in the numerator; that is, it is a polynomial in v. Hence, the ith coefficient of r with respect to v is well-defined, and indeed we can define the result very similarly: that calling sequence returns the same result as UnivariatePolynomialOverPowerSeries(lp, v) where lp := [seq(PowerSeries(coeff(r,v,i)),i=0..degree(r,v))] or lp := [seq(PowerSeries(coeff(p,v,i)),i=0..degree(p,v))].

The calling sequence UnivariatePolynomialOverPowerSeries(ps, v) creates a univariate polynomial over power series representing ps, with v as its main variable. This is only possible if ps is known to be a polynomial function of v, which is the case if it is independent of v (in which case it is trivially polynomial) or if the analytic expression for ps is known and it is polynomial in v. If neither of the former two cases is true, then an error is raised.

The calling sequence UnivariatePolynomialOverPowerSeries(u, v) copies u. You may omit v in this case, but if you specify it, it must be equal to the main variable of u.

The calling sequences with UnivariatePolynomialOverPuiseuxSeries are exactly the same as their equivalents using UnivariatePolynomialOverPowerSeries; the underlying data structure supports both power series and Puiseux series as the coefficients. The alias UnivariatePolynomialOverPuiseuxSeries was added to clarify the fact that the coefficients can be 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.

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

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1\right)\,\mathrm{PowerSeries}\left(0\right)\,\mathrm{PowerSeries}\left(x\right)\,\mathrm{PowerSeries}\left(y\right)\,\frac{1}{\mathrm{PowerSeries}\left(1+x+y\right)}\right]\,z\right)$

$f≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(0\right)z\mathrm{+}\left(x\right)z^2\mathrm{+}\left(y\right)z^3\mathrm{+}\left(1+\mathrm{\dots }\right)z^4\right]$

$\mathrm{Degree}\left(f\right)$

$4$

$\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(f\,i\right)\,i=0..4\right)\right]$

$\left[\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}0\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}x\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}y\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1+x+y}\mathrm{:}1+\mathrm{\dots }\right]\right]$

$\mathrm{Truncate}\left(f\,1\right)$

$1+\left(1-x-y\right)z^4+y{z}^3+x{z}^2$

$\mathrm{Truncate}\left(f\,2\right)$

$1+\left(x^2+2xy+y^2-x-y+1\right)z^4+y{z}^3+x{z}^2$

$p≔\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left(z^2+z\right)$

$p≔\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left(z^2+z\right)$

$\mathrm{UnivariatePolynomialOverPowerSeries}\left(p\,z\right)$

$\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(\mathrm{-6}\right)\mathrm{+}\left(11+x\right)z\mathrm{+}\left(\mathrm{-6}+x\right)z^2\mathrm{+}\left(1\right)z^3\right]$

$r≔\frac{p}{x+y-1}+1$

$r≔\frac{\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left(z^2+z\right)}{x+y-1}+1$

$\mathrm{UnivariatePolynomialOverPowerSeries}\left(r\,z\right)$

$\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(7+\mathrm{\dots }\right)\mathrm{+}\left(\mathrm{-11}+\mathrm{\dots }\right)z\mathrm{+}\left(6+\mathrm{\dots }\right)z^2\mathrm{+}\left(\mathrm{-1}+\mathrm{\dots }\right)z^3\right]$

$p≔\frac{1+x^{2}z+z^2}{x^2+x}$

$p≔\frac{x^{2}z+z^2+1}{x^2+x}$

$h≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(p\,z\right)$

$h≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(\frac{1}{x}+\mathrm{\dots }\right)\mathrm{+}\left(0+\mathrm{\dots }\right)z\mathrm{+}\left(\frac{1}{x}+\mathrm{\dots }\right)z^2\right]$

$\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(h\,i\right)\,i=0..2\right)\right]$

$\left[\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{1}{x\left(x+1\right)}\mathrm{:}\frac{1}{x}+\mathrm{\dots }\right]\,\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{x}{x+1}\mathrm{:}0+\mathrm{\dots }\right]\,\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{1}{x\left(x+1\right)}\mathrm{:}\frac{1}{x}+\mathrm{\dots }\right]\right]$

$d≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d=0\,0\,\frac{z\cdot x^{d-1}}{\left(d-1\right)!}\right)\,\mathrm{variables}=\left\{x\,z\right\}\right)$

$d≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}0+\mathrm{\dots }\right]$

$\mathrm{UnivariatePolynomialOverPowerSeries}\left(d\,z\right)$

Error, (in MultivariatePowerSeries:-UnivariatePolynomialOverPowerSeries) attempted to convert a power series involving z to a univariate polynomial over power series in z, but it is not known to be polynomial in z

$e≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d=0\,0\,\frac{z\cdot x^{d-1}}{\left(d-1\right)!}\right)\,\mathrm{analytic}=z\exp \left(x\right)\right)$

$e≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}z{\ⅇ}^x\mathrm{:}0+\mathrm{\dots }\right]$

$k≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(e\,z\right)$

$k≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\right]$

$\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\,x\right)$

Error, (in MultivariatePowerSeries:-UnivariatePolynomialOverPowerSeries) you specified x as the main variable, but the main variable of the first argument is z

$\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\,z\right)$

$\mathrm{k1}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\right]$

$\mathrm{k2}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\right)$

$\mathrm{k2}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\right]$

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

$g≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(0\right)z\mathrm{+}\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)z^2\mathrm{+}\left(\sqrt{y}\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\right]$

$\mathrm{Degree}\left(g\right)$

$4$

$\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(g\,i\right)\,i=0..4\right)\right]$

$\left[\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}1\right]\,\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}0\right]\,\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}{x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{:}{x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]\,\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\sqrt{y}\mathrm{:}\sqrt{y}\right]\,\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{x\sqrt{y}+\frac{x}{y}}{1+x\sqrt{y}+\frac{x}{y}}\mathrm{:}0+\mathrm{\dots }\right]\right]$

$\mathrm{Truncate}\left(g\,1\right)$

$1+\left(x\sqrt{y}+\frac{x}{y}\right)z^4+\sqrt{y}{z}^3+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{z}^2$

$\mathrm{Truncate}\left(g\,2\right)$

$1+\left(-x^{2}y-\frac{2x^2}{\sqrt{y}}-\frac{x^2}{y^2}+x\sqrt{y}+\frac{x}{y}\right)z^4+\sqrt{y}{z}^3+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{z}^2$

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

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

The r parameter was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.

The MultivariatePowerSeries[UnivariatePolynomialOverPuiseuxSeries] command was introduced in Maple 2023.

The MultivariatePowerSeries[UnivariatePolynomialOverPowerSeries] command was updated in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.