 UnivariatePolynomialOverPowerSeries - Maple Help

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MultivariatePowerSeries

 UnivariatePolynomialOverPowerSeries
 create a univariate polynomial over power series or Puiseux series
 UnivariatePolynomialOverPuiseuxSeries
 create a univariate polynomial over power series or Puiseux series Calling Sequence

 UnivariatePolynomialOverPowerSeries(lp, v) UnivariatePolynomialOverPuiseuxSeries(lp, v) UnivariatePolynomialOverPowerSeries(p, v) UnivariatePolynomialOverPuiseuxSeries(p, v) UnivariatePolynomialOverPowerSeries(r, v) UnivariatePolynomialOverPuiseuxSeries(r, v) UnivariatePolynomialOverPowerSeries(ps, v) UnivariatePolynomialOverPuiseuxSeries(ps, v) UnivariatePolynomialOverPowerSeries(u) UnivariatePolynomialOverPuiseuxSeries(u) UnivariatePolynomialOverPowerSeries(u, v) UnivariatePolynomialOverPuiseuxSeries(u, v) Parameters

 lp - list, Array, or Vector of power series or Puiseux series generated by this package v - variable p - polynomial r - rational function where v does not occur in the denominator ps - power series generated by this package u - univariate polynomial over power series generated by this package Description

 • 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. Examples

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

Create a univariate polynomial over power series from a list of power series.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1\right),\mathrm{PowerSeries}\left(0\right),\mathrm{PowerSeries}\left(x\right),\mathrm{PowerSeries}\left(y\right),\frac{1}{\mathrm{PowerSeries}\left(1+x+y\right)}\right],z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({x}\right){}{{z}}^{{2}}{+}\left({y}\right){}{{z}}^{{3}}{+}\left({1}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (1)
 > $\mathrm{Degree}\left(f\right)$
 ${4}$ (2)
 > $\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(f,i\right),i=0..4\right)\right]$
 $\left[\left[{PowⅇrSⅇriⅇs:}{1}\right]{,}\left[{PowⅇrSⅇriⅇs:}{0}\right]{,}\left[{PowⅇrSⅇriⅇs:}{x}\right]{,}\left[{PowⅇrSⅇriⅇs:}{y}\right]{,}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{+}{x}{+}{y}}{:}{1}{+}{\dots }\right]\right]$ (3)

We compute its linear and quadratic truncation. These are defined in terms of the homogeneous degree of the coefficients, ignoring the degree in the main variable.

 > $\mathrm{Truncate}\left(f,1\right)$
 ${1}{+}\left({1}{-}{x}{-}{y}\right){}{{z}}^{{4}}{+}{y}{}{{z}}^{{3}}{+}{x}{}{{z}}^{{2}}$ (4)
 > $\mathrm{Truncate}\left(f,2\right)$
 ${1}{+}\left({{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{-}{x}{-}{y}{+}{1}\right){}{{z}}^{{4}}{+}{y}{}{{z}}^{{3}}{+}{x}{}{{z}}^{{2}}$ (5)

Create a univariate polynomial over power series from a polynomial and a rational function.

 > $p≔\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left({z}^{2}+z\right)$
 ${p}{≔}\left({z}{-}{1}\right){}\left({z}{-}{2}\right){}\left({z}{-}{3}\right){+}{x}{}\left({{z}}^{{2}}{+}{z}\right)$ (6)
 > $\mathrm{UnivariatePolynomialOverPowerSeries}\left(p,z\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({-6}\right){+}\left({11}{+}{x}\right){}{z}{+}\left({-6}{+}{x}\right){}{{z}}^{{2}}{+}\left({1}\right){}{{z}}^{{3}}\right]$ (7)
 > $r≔\frac{p}{x+y-1}+1$
 ${r}{≔}\frac{\left({z}{-}{1}\right){}\left({z}{-}{2}\right){}\left({z}{-}{3}\right){+}{x}{}\left({{z}}^{{2}}{+}{z}\right)}{{x}{+}{y}{-}{1}}{+}{1}$ (8)
 > $\mathrm{UnivariatePolynomialOverPowerSeries}\left(r,z\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({7}{+}{\dots }\right){+}\left({-11}{+}{\dots }\right){}{z}{+}\left({6}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({-1}{+}{\dots }\right){}{{z}}^{{3}}\right]$ (9)

Notice that the following example returns a univariate polynomial with Puiseux series as coefficients.

 > $p≔\frac{1+{x}^{2}z+{z}^{2}}{{x}^{2}+x}$
 ${p}{≔}\frac{{{x}}^{{2}}{}{z}{+}{{z}}^{{2}}{+}{1}}{{{x}}^{{2}}{+}{x}}$ (10)
 > $h≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(p,z\right)$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left(\frac{{1}}{{x}}{+}{\dots }\right){+}\left({0}{+}{\dots }\right){}{z}{+}\left(\frac{{1}}{{x}}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (11)
 > $\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(h,i\right),i=0..2\right)\right]$
 $\left[\left[{PuisⅇuxSⅇriⅇs of}\frac{{1}}{{x}{}\left({x}{+}{1}\right)}{:}\frac{{1}}{{x}}{+}{\dots }\right]{,}\left[{PuisⅇuxSⅇriⅇs of}\frac{{x}}{{x}{+}{1}}{:}{0}{+}{\dots }\right]{,}\left[{PuisⅇuxSⅇriⅇs of}\frac{{1}}{{x}{}\left({x}{+}{1}\right)}{:}\frac{{1}}{{x}}{+}{\dots }\right]\right]$ (12)

This happens since the polynomial x^2+x is not invertible as a power series, but it is as a Puiseux series.

The following attempt will not work, because Maple cannot determine that d is polynomial in z (though actually it is).

 > $d≔\mathrm{PowerSeries}\left(d→\mathrm{ifelse}\left(d=0,0,\frac{z{x}^{d-1}}{\left(d-1\right)!}\right),\mathrm{variables}=\left\{x,z\right\}\right)$
 ${d}{≔}\left[{PowⅇrSⅇriⅇs:}{0}{+}{\dots }\right]$ (13)
 > $\mathrm{UnivariatePolynomialOverPowerSeries}\left(d,z\right)$

We define e in the same way as d but specify the analytic expression. Then we can successfully convert it to a univariate polynomial over power series.

 > $e≔\mathrm{PowerSeries}\left(d→\mathrm{ifelse}\left(d=0,0,\frac{z{x}^{d-1}}{\left(d-1\right)!}\right),\mathrm{analytic}=z{ⅇ}^{x}\right)$
 ${e}{≔}\left[{PowⅇrSⅇriⅇs of}{z}{}{{ⅇ}}^{{x}}{:}{0}{+}{\dots }\right]$ (14)
 > $k≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(e,z\right)$
 ${k}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{+}{\dots }\right){}{z}\right]$ (15)

To copy k, we can specify the main variable explicitly or omit it. If we specify it explicitly, it has to be z, otherwise we obtain an error:

 > $\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k,x\right)$
 > $\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k,z\right)$
 ${\mathrm{k1}}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{+}{\dots }\right){}{z}\right]$ (16)
 > $\mathrm{k2}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\right)$
 ${\mathrm{k2}}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{+}{\dots }\right){}{z}\right]$ (17)

Create a univariate polynomial over Puiseux series from a list of Puiseux series.

 > $g≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (18)
 > $\mathrm{Degree}\left(g\right)$
 ${4}$ (19)
 > $\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(g,i\right),i=0..4\right)\right]$
 $\left[\left[{PuisⅇuxSⅇriⅇs:}{1}\right]{,}\left[{PuisⅇuxSⅇriⅇs:}{0}\right]{,}\left[{PuisⅇuxSⅇriⅇs of}{{x}}^{{1}}{{3}}}{:}{{x}}^{{1}}{{3}}}\right]{,}\left[{PuisⅇuxSⅇriⅇs of}\sqrt{{y}}{:}\sqrt{{y}}\right]{,}\left[{PuisⅇuxSⅇriⅇs of}\frac{{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{{1}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{:}{0}{+}{\dots }\right]\right]$ (20)

We compute its linear and quadratic truncation. These are defined in terms of the homogeneous degree of the internal power series objects in $g$s coefficients, ignoring the degree in the main variable.

 > $\mathrm{Truncate}\left(g,1\right)$
 ${1}{+}\left({x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}\right){}{{z}}^{{4}}{+}\sqrt{{y}}{}{{z}}^{{3}}{+}{{x}}^{{1}}{{3}}}{}{{z}}^{{2}}$ (21)
 > $\mathrm{Truncate}\left(g,2\right)$
 ${1}{+}\left({-}{{x}}^{{2}}{}{y}{-}\frac{{2}{}{{x}}^{{2}}}{\sqrt{{y}}}{-}\frac{{{x}}^{{2}}}{{{y}}^{{2}}}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}\right){}{{z}}^{{4}}{+}\sqrt{{y}}{}{{z}}^{{3}}{+}{{x}}^{{1}}{{3}}}{}{{z}}^{{2}}$ (22) Compatibility

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