Degree - Maple Help

MultivariatePowerSeries

 Degree
 degree of a univariate polynomial over power series or over Puiseux series

 Calling Sequence Degree(u)

Parameters

 u - univariate polynomial over power series or over Puiseux series generated by this package

Description

 • Degree(u) returns the syntactic degree of the univariate polynomial over power series  or over Puiseux series u with respect to its main variable. That is, it does not test whether the highest degree coefficient is nonzero (because this cannot be done in general); it just returns the highest degree coefficient of the main variable that is specified.
 • 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

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

This univariate polynomial over power series is quadratic in its main variable, $z$.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(1+{x}^{3}y+xyz+\left(1+y\right){z}^{2},z\right):$
 > $\mathrm{Degree}\left(f\right)$
 ${2}$ (1)

This univariate polynomial over power series is syntactically cubic in its main variable, $u$, but the coefficient of ${u}^{3}$ is 0.

 > $g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(x\right),\mathrm{PowerSeries}\left(y\right),\mathrm{GeometricSeries}\left(\left[x\right]\right),\mathrm{PowerSeries}\left(0\right)\right],u\right):$
 > $\mathrm{Degree}\left(g\right)$
 ${3}$ (2)

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

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

We verify the degree of h with the Degree command.

 > $\mathrm{Degree}\left(h\right)$
 ${4}$ (4)

Compatibility

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