These commands set two parameters in a Puiseux series object. The parameters, called the nonzero_pso_bound for SetNonzeroPowerSeriesDegreeBound and smallest_term_bound for SetSmallestTermDegreeBound, are used in computing multiplicative inverses. The function of the parameters is explained on the Inverse help page.

Using the first or third calling sequence, we set the nonzero_pso_bound or smallest_term_bound, respectively, equal to bound for the given object s. This is used when computing the inverse of s (unless overridden in the Inverse command itself).

Using the second or fourth calling sequence, we set the global default value for nonzero_pso_bound or smallest_term_bound, respectively. This value is used when computing the inverse of a Puiseux series object for which that parameter has not been set using the first or third calling sequence (unless overridden in the Inverse command itself). Initially, the default value for both parameters is 10.

All calling sequences return the previously stored value for the given parameter.

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.

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

$p≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{piecewise}\left(d=0\,1\,d=1\,u+v\,0\right)\,\mathrm{variables}=\left\{u\,v\right\}\right)$

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

$X≔\left[x\,y\right]\;$$U≔\left[u\,v\right]$

$X≔\left[x\,y\right]$

$U≔\left[u\,v\right]$

$\mathrm{R1}≔\left[\left[1\,0\right]\,\left[1\,-1\right]\right]\;$$\mathrm{R2}≔\left[\left[2\,2\right]\,\left[2\,-2\right]\right]$

$\mathrm{R1}≔\left[\left[1\,0\right]\,\left[1\,\mathrm{-1}\right]\right]$

$\mathrm{R2}≔\left[\left[2\,2\right]\,\left[2\,\mathrm{-2}\right]\right]$

$\mathrm{E1}≔\left[x=-5\,y=3\right]\;$$\mathrm{E2}≔\left[x=12\,y=8\right]$

$\mathrm{E1}≔\left[x=\mathrm{-5}\,y=3\right]$

$\mathrm{E2}≔\left[x=12\,y=8\right]$

$s≔\mathrm{PuiseuxSeries}\left(p\,X\,U\,\mathrm{R1}\,\mathrm{E1}\right)$

$s≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}\frac{y^3}{x^5}+\mathrm{\dots }\right]$

$t≔\mathrm{PuiseuxSeries}\left(p\,X\,U\,\mathrm{R2}\,\mathrm{E2}\right)$

$t≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}{x}^{12}{y}^8+\mathrm{\dots }\right]$

$\mathrm{SetNonzeroPowerSeriesDegreeBound}\left(15\,s\right)$

$\mathrm{undefined}$

$\mathrm{SetNonzeroPowerSeriesDegreeBound}\left(17\right)$

$10$

$\mathrm{SetSmallestTermDegreeBound}\left(5\,t\right)$

$\mathrm{undefined}$

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

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}\frac{x^5}{y^3}+\mathrm{\dots }\right]$

$\mathrm{Inverse}\left(t\right)$

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs:}\frac{1}{x^{12}{y}^8}+\mathrm{\dots }\right]$

Monforte, A.A., & Kauers, M. "Formal Laurent series in several variables." Expositiones Mathematicae. Vol. 31 No. 4 (2013): 350-367.

The MultivariatePowerSeries[SetNonzeroPowerSeriesDegreeBound] and MultivariatePowerSeries[SetSmallestTermDegreeBound] commands were introduced in Maple 2023.

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