TschirnhausenTransformation - Maple Help

MultivariatePowerSeries

 TschirnhausenTransformation
 apply a linear transformation to a univariate polynomial over power series with Puiseux coefficients

 Calling Sequence TschirnhausenTransformation(u, a) TschirnhausenTransformation(u, a, m)

Parameters

 u - univariate polynomial over power series with Puiseux coefficients in one variable generated by this package a - Puiseux series generated by this package m - (optional) non-negative integer

Description

 • 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.

Examples

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

We create a univariate polynomial over power series from a list of Puiseux series.

 > $a≔\mathrm{PuiseuxSeries}\left(\frac{x}{1+x},\left[x={x}^{\frac{1}{2}}\right]\right)$
 ${a}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{\sqrt{{x}}}{{1}{+}\sqrt{{x}}}{:}{0}{+}{\dots }\right]$ (1)
 > $f≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{2}}\right]\right),a,\mathrm{PuiseuxSeries}\left(1\right)\right],z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{x}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}{+}\left({1}\right){}{{z}}^{{5}}\right]$ (2)

We apply the transformation $z-\frac{a}{5}$ to f.

 > $g≔\mathrm{TschirnhausenTransformation}\left(f,a\right)$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}{+}{\dots }\right){+}\left({0}{+}{\dots }\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({0}\right){}{{z}}^{{4}}{+}\left({1}\right){}{{z}}^{{5}}\right]$ (3)

Observe that the monomial of degree 4 is 0.

 > $\mathrm{GetCoefficient}\left(g,4\right)$
 $\left[{PuisⅇuxSⅇriⅇs:}{0}\right]$ (4)

Finally, observe that by applying the linear transformation $z+\frac{a}{5}$ to g, we get f back.

 > $h≔\mathrm{TschirnhausenTransformation}\left(g,\mathrm{Negate}\left(a\right),5\right)$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}{+}{\dots }\right){+}\left({0}{+}{\dots }\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}{+}\left({1}\right){}{{z}}^{{5}}\right]$ (5)
 > $\mathrm{Truncate}\left(\mathrm{Subtract}\left(f,h\right),10\right)$
 ${0}$ (6)

Compatibility

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