This command sets a parameter in a univariate polynomial over power series . The parameter is used in computing the extended Hensel construction. Its function is described on the ExtendedHenselConstruction help page.

Using the first calling sequence, we set the bound for the given object u. This is used when computing the extended Hensel construction of u (unless overridden in the ExtendedHenselConstruction 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 for which the bound has not been set using the first calling sequence (unless overridden in the ExtendedHenselConstruction 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.

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

bproc := proc(d) if d  = 10 then return y^(10);    else return 0; end if; end proc;

$\mathrm{bproc}≔\mathbf{proc}\left(d\right)\mathbf{if}d\=10\mathbf{then}\mathbf{return}y\^10\mathbf{else}\mathbf{return}0\mathbf{end\; if}\mathbf{end\; proc}$

$\mathrm{pso}≔\mathrm{PowerSeries}\left(\mathrm{bproc}\,\mathrm{variables}=\left\{y\right\}\right)$

$\mathrm{pso}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}0+\mathrm{\dots }\right]$

bproc1 := proc(d) if d  = 4 then return y^(4); else return 0; end if; end proc;

$\mathrm{bproc1}≔\mathbf{proc}\left(d\right)\mathbf{if}d\=4\mathbf{then}\mathbf{return}y\^4\mathbf{else}\mathbf{return}0\mathbf{end\; if}\mathbf{end\; proc}$

$\mathrm{pso1}≔\mathrm{PowerSeries}\left(\mathrm{bproc1}\,\mathrm{variables}=\left\{y\right\}\right)$

$\mathrm{pso1}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}0+\mathrm{\dots }\right]$

bproc2 := proc(d) if d  = 20 then return y^(20); else return 0; end if; end proc;

$\mathrm{bproc2}≔\mathbf{proc}\left(d\right)\mathbf{if}d\=20\mathbf{then}\mathbf{return}y\^20\mathbf{else}\mathbf{return}0\mathbf{end\; if}\mathbf{end\; proc}$

$\mathrm{pso2}≔\mathrm{PowerSeries}\left(\mathrm{bproc2}\,\mathrm{variables}=\left\{y\right\}\right)$

$\mathrm{pso2}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}0+\mathrm{\dots }\right]$

$u≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{pso}\,\mathrm{pso1}\,\mathrm{pso2}\,\mathrm{pso}\right]\,x\right)$

$u≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0+\mathrm{\dots }\right)\mathrm{+}\left(0+\mathrm{\dots }\right)x\mathrm{+}\left(0+\mathrm{\dots }\right)x^2\mathrm{+}\left(0+\mathrm{\dots }\right)x^3\right]$

$\mathrm{ExtendedHenselConstruction}\left(u\right)$

Error, (in MultivariatePowerSeries:-ExtendedHenselConstruction) it is not possible to compute the Newton line with the current bound 10, try to increase it

$\mathrm{SetHenselBound}\left(20\right)$

$10$

$\mathrm{ExtendedHenselConstruction}\left(u\right)$

$\left[\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(y^{10}+\mathrm{\dots }\right)\right]\,\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(0+\mathrm{\dots }\right)\mathrm{+}\left(1+\mathrm{\dots }\right)x\right]\,\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(0+\mathrm{\dots }\right)\mathrm{+}\left(1+\mathrm{\dots }\right)x\right]\,\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(0+\mathrm{\dots }\right)\mathrm{+}\left(1+\mathrm{\dots }\right)x\right]\right]$

$\mathrm{SetHenselBound}\left(15\,u\right)$

$\mathrm{undefined}$

$\mathrm{ExtendedHenselConstruction}\left(u\right)$

$\left[\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(y^{10}+\mathrm{\dots }\right)\right]\,\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(0+\mathrm{\dots }\right)\mathrm{+}\left(1+\mathrm{\dots }\right)x\right]\,\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(0+\mathrm{\dots }\right)\mathrm{+}\left(1+\mathrm{\dots }\right)x\right]\,\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(0+\mathrm{\dots }\right)\mathrm{+}\left(1+\mathrm{\dots }\right)x\right]\right]$

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

Error, (in MultivariatePowerSeries:-ExtendedHenselConstruction) leading coefficient of Object(MultivariatePowerSeries:-UnivariatePolynomialOverPowerSeriesObject,Array(0..3, [Object(MultivariatePowerSeries:-PowerSeriesObject,Array(0..21, [0,0,0,0,0,0,0,0,0,0,y^10,0,0,0,0,0,0,0,0,0,0,0]),21,MultivariatePowerSeries:-PowerSeriesObject:-proc_gen,{y},ancestors),Object(MultivariatePowerSeries:-PowerSeriesObject,Array(0..20, [0,0,0,0,y^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]),20,MultivariatePowerSeries:-PowerSeriesObject:-proc_gen,{y},ancestors),Object(MultivariatePowerSeries:-PowerSeriesObject,Array(0..20, [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,y^20]),20,MultivariatePowerSeries:-PowerSeriesObject:-proc_gen,{y},ancestors),Object(MultivariatePowerSeries:-PowerSeriesObject,Array(0..21, [0,0,0,0,0,0,0,0,0,0,y^10,0,0,0,0,0,0,0,0,0,0,0]),21,MultivariatePowerSeries:-PowerSeriesObject:-proc_gen,{y},ancestors)])) may be zero

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

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

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