SetPuiseuxBound - Maple Help
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MultivariatePowerSeries

 SetPuiseuxBound
 set a Puiseux_bound for a univariate polynomial over power series

 Calling Sequence SetPuiseuxBound(bound, u) SetPuiseuxBound(bound)

Parameters

 bound - a non-negative integer u - (optional) univariate polynomial over power series or over Puiseux series generated by this package

Description

 • This command sets a parameter in a univariate polynomial over power series or over Puiseux series. The parameter is used in computing the Puiseux factorization. Its function is described on the PuiseuxFactorize help page; it involves a call to the GetOrder command.
 • Using the first calling sequence, we set the bound for the given object u. This is used when computing the Puiseux factorization of u (unless overridden in the PuiseuxFactorize command itself).
 • Using the second calling sequence, we set the global default value for the bound. This value is used when computing the Puiseux factorization of a univariate polynomial over power series or over Puiseux series for which the bound has not been set using the first calling sequence (unless overridden in the PuiseuxFactorize command itself). Initially, the default value for the bound is 10.
 • Both calling sequences return the previously stored value for the bound, either in u (for the first calling sequence) or the default (for the second).
 • 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 Puiseux series.

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

Computing the Puiseux factorization now would use the initial value of the global default bound, 10. This is too low to determine the order of one of the intermediate expressions in the computation (it needs to be 12), so we see an error message.

 > $\mathrm{PuiseuxFactorize}\left(u\right)$

If we set a different global default, that's the value that is used. This value is high enough.

 > $\mathrm{SetPuiseuxBound}\left(17\right)$
 ${10}$ (2)
 > $\mathrm{PuiseuxFactorize}\left(u\right)$
 $\left[\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right)\right]{,}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left(\frac{{1}}{{{x}}^{{2}}}\right)\right]{,}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({0}{+}{\dots }\right){+}\left(\sqrt{{x}}\right){}{z}\right]{,}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({0}{+}{\dots }\right){+}\left({0}{+}{\dots }\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({{x}}^{{3}}{{2}}}\right){}{{z}}^{{3}}\right]\right]$ (3)

If we set the bound for u specifically, that overrides the default value.

 > $\mathrm{SetPuiseuxBound}\left(15,u\right)$
 ${\mathrm{undefined}}$ (4)
 > $\mathrm{PuiseuxFactorize}\left(u\right)$
 $\left[\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right)\right]{,}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left(\frac{{1}}{{{x}}^{{2}}}\right)\right]{,}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({0}{+}{\dots }\right){+}\left(\sqrt{{x}}\right){}{z}\right]{,}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({0}{+}{\dots }\right){+}\left({0}{+}{\dots }\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({{x}}^{{3}}{{2}}}\right){}{{z}}^{{3}}\right]\right]$ (5)

If we specify the bound in the PuiseuxFactorize command itself, that overrides all set values. This is, once again, too low of a value for the bound.

 > $\mathrm{PuiseuxFactorize}\left(u,8\right)$

References

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

Compatibility

 • The MultivariatePowerSeries[SetPuiseuxBound] command was introduced in Maple 2023.
 • For more information on Maple 2023 changes, see Updates in Maple 2023.

 See Also