The GetOrder command computes the order of a Puiseux series in one variable, s.

A univariate Puiseux series can be written as $\sum _{m=M}^{\mathrm{\infty }}{a}_m^{\frac{m}{n}}$ for some $M\in \mathbb{Z}$, $n\in \mathbb{N}$. Then the order of s is defined as $\min \left(\left\{\frac{m}{n}\|a_m\ne 0\right\}\right)$.

If the order cannot be determined from the analytic expression for s, the command examines subsequent homogeneous components from the internal power series of s. If it has not found any nonzero components after examining bnd + 1 such components (degrees 0 through bnd), the command gives up and signals an error. The default value for bnd is 10.

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.

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

$p≔\mathrm{PowerSeries}\left(d\mapsto {\left(-u\right)}^d\,\mathrm{variables}=\left\{u\right\}\right)$

$p≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+\mathrm{\dots }\right]$

$s≔\mathrm{PuiseuxSeries}\left(p\,\left[u=x^2\right]\right)$

$s≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}1+\mathrm{\dots }\right]$

$\mathrm{s2}≔\mathrm{PuiseuxSeries}\left(p\,\left[u=x^{\frac{2}{3}}\right]\right)-1$

$\mathrm{s2}≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}0+\mathrm{\dots }\right]$

$\mathrm{GetOrder}\left(s\right)$

$0$

$\mathrm{GetOrder}\left(\mathrm{s2}\right)$

$\frac{2}{3}$

$\mathrm{GetOrder}\left(s-1\right)$

$2$

$\mathrm{GetOrder}\left(s-\left(1-x^2\right)\right)$

$4$

$\mathrm{GetOrder}\left({\mathrm{s2}}^{12}\right)$

Error, (in MultivariatePowerSeries:-GetOrder) the order of Object(MultivariatePowerSeries:-PuiseuxSeriesObject,Object(MultivariatePowerSeries:-PowerSeriesObject,Array(0..10, [0,0,0,0,0,0,0,0,0,0,0]),10,MultivariatePowerSeries:-PowerSeriesObject:-mul_gen,{u},ancestors),[x],[u],[[2/3]],[x = 0]) cannot be determined with the current bound 10

$\mathrm{GetOrder}\left({\mathrm{s2}}^{12}\,15\right)$

$8$

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

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