MultivariatePowerSeries/TaylorShift - Maple Help

MultivariatePowerSeries

 TaylorShift
 Perform a Taylor shift of a univariate polynomial over power series

 Calling Sequence TaylorShift(u, c)

Parameters

 u - univariate polynomial over power series generated by this package c - numeric value or algebraic number

Description

 • The command TaylorShift(u, c) returns the univariate polynomial over power series obtained by substituting v + c for v in u, where v is the main variable of u. In other words, the result is obtained by composing u with the map that sends v to v + c.
 • A typical usage is when c is a root of the polynomial returned by EvaluateAtOrigin(u). This happens, for example, in HenselFactorize.
 • 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.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1\right),\mathrm{SumOfAllMonomials}\left(\left[x,y\right]\right),\mathrm{GeometricSeries}\left(y\right)\right],z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}\right){+}\left({1}{+}{x}{+}{y}{+}{\dots }\right){}{z}{+}\left({1}{+}{y}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (1)

We apply a Taylor shift by 1, and then by -1 on the result.

 > $\mathrm{f1}≔\mathrm{TaylorShift}\left(f,1\right)$
 ${\mathrm{f1}}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({3}{+}{\dots }\right){+}\left({3}{+}{x}{+}{3}{}{y}{+}{\dots }\right){}{z}{+}\left({1}{+}{y}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (2)
 > $\mathrm{f0}≔\mathrm{TaylorShift}\left(\mathrm{f1},-1\right)$
 ${\mathrm{f0}}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}\right){+}\left({1}{+}{x}{+}{y}{+}{\dots }\right){}{z}{+}\left({1}{+}{y}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (3)

We verify that the result is equal to the original polynomial (up to homogeneous degree 20).

 > $\mathrm{ApproximatelyEqual}\left(f,\mathrm{f0},20\right)$
 ${\mathrm{true}}$ (4)

Compatibility

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