The command TschirnhausenTransformation(u, a) applies the linear transformation $x-\frac{a}{n}$ to u, where n is the degree of a as a polynomial in the variable x. This transformation is called the Tschirnhausen transformation or the Tschirnhaus transformation. If u is monic, then the output of this command produces a univariate polynomial over power series with Puiseux series coefficients such that the monomial of degree n-1 is equal to zero.

The command TschirnhausenTransformation(u, a, m) applies the linear transformation $x-\frac{a}{m}$ to u, with a a univariate polynomial in the variable x.

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.

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

$a≔\mathrm{PuiseuxSeries}\left(\frac{x}{1+x}\,\left[x=x^{\frac{1}{2}}\right]\right)$

$a≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{\sqrt{x}}{1+\sqrt{x}}\mathrm{:}0+\mathrm{\dots }\right]$

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

$f≔\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{x}\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\mathrm{+}\left(1\right)z^5\right]$

$g≔\mathrm{TschirnhausenTransformation}\left(f\,a\right)$

$g≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(1+\mathrm{\dots }\right)\mathrm{+}\left(0+\mathrm{\dots }\right)z\mathrm{+}\left(0+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\mathrm{+}\left(0\right)z^4\mathrm{+}\left(1\right)z^5\right]$

$\mathrm{GetCoefficient}\left(g\,4\right)$

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}0\right]$

$h≔\mathrm{TschirnhausenTransformation}\left(g\,\mathrm{Negate}\left(a\right)\,5\right)$

$h≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(1+\mathrm{\dots }\right)\mathrm{+}\left(0+\mathrm{\dots }\right)z\mathrm{+}\left(0+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\mathrm{+}\left(1\right)z^5\right]$

$\mathrm{Truncate}\left(\mathrm{Subtract}\left(f\,h\right)\,10\right)$

$0$

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

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