PuiseuxSeries - Maple Help

MultivariatePowerSeries

 PuiseuxSeries
 create a Puiseux series

 Calling Sequence PuiseuxSeries(g, X, U, R, e) PuiseuxSeries(g, mp, e)

Parameters

 g - polynomial, rational function, or power series generated by this package X - (optional) list of ordered variables for the Puiseux series U - (optional) list of ordered variables for the power series R - (optional) list of grevlex positive rational rays e - (optional) list of equations representing the exponents of a monomial multiplying the Puiseux series mp - list of equations representing the change of variables to be applied to g

Description

 • The PuiseuxSeries command is used to create an object representing a Puiseux series.
 • A Puiseux series is a power series in rational powers of the variables. More precisely:
 – Let $X≔\left({x}_{1},\dots ,{x}_{p}\right)$ and $U≔\left({u}_{1},\dots ,{u}_{m}\right)$ be ordered lists of variables.
 – Let $R≔\left({r}_{1},\dots ,{r}_{m}\right)$ be a list of $m$ grevlex-positive $p$-dimensional rational vectors.
 – Let $e≔\left({e}_{1},\dots ,{e}_{p}\right)$ be a point in ${\mathrm{ℚ}}^{p}$.
 – Let $g\left(U\right)≔{\sum }_{n=0}^{\mathrm{\infty }}{g}_{n}\left(U\right)$ be a multivariate power series in $U$ with homogeneous components ${g}_{n}\left(U\right)$.
 For any $v=\left({v}_{1},\dots ,{v}_{q}\right)$ in ${\mathrm{ℚ}}^{q}$ and any list $Y=\left({y}_{1},\dots ,{y}_{q}\right)$, we write ${Y}^{v}$ for ${y}_{1}^{{v}_{1}}\dots {y}_{q}^{{v}_{q}}$. Moreover, we write ${X}^{R}$ for the list $\left({X}^{{r}_{1}},\dots ,{X}^{{r}_{m}}\right)$ of $m$ products of powers of the variables in $X$. Then $P≔{X}^{e}g\left({X}^{R}\right)$ is a Puiseux series, and every Puiseux series can be written in this way. This can be understood as evaluating $g\left(U\right)$ at ${u}_{i}={X}^{{r}_{i}}$ and then multiplying the result by ${X}^{e}$.
 • We call $g$ the internal power series of the Puiseux series $P$; $X$ the variable order of $P$; $U$ the variable order of $g$; and $R$ the rays of $P$. The rays generate the cone containing the support of $P$, meaning the set of exponent vectors of $X$ that occur in $P$ with a nonzero coefficient, as in the paper by Monforte and Kauers (see References). The vertex of this cone is $e$.
 • The calling sequence PuiseuxSeries(g, X, U, R, E) creates an object representing $P$, where:
 – g is a polynomial in $X$, or a formal multivariate power series in ${\mathrm{ℂ}}_{C}⟦X⟧$,
 – R is a list of grevlex positive $p$-dimensional rays contained in ${\mathrm{ℚ}}^{p}$,
 – E is a list of the form $\left[{x}_{1}={e}_{1},\dots ,{x}_{p}={e}_{p}\right]$ with $e=\left({e}_{1},\dots ,{e}_{p}\right)$ in ${\mathrm{ℚ}}^{p}$.
 • The calling sequence PuiseuxSeries(g, mp, e) creates an object representing a Puiseux series obtained by substituting the equations in mp into g. The list mp must have one equation for each of the variables in g.
 • 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 Puiseux series, determine its inverse, multiply them and find its truncation to homogeneous degree 15.

 > $p≔\mathrm{PowerSeries}\left(1+uv\right);$$X≔\left[x,y\right];$$U≔\left[u,v\right];$$R≔\left[\left[1,0\right],\left[1,-\frac{1}{2}\right]\right];$$E≔\left[x=-5,y=3\right]$
 ${p}{≔}\left[{PowⅇrSⅇriⅇs:}{1}{+}{u}{}{v}\right]$
 ${X}{≔}\left[{x}{,}{y}\right]$
 ${U}{≔}\left[{u}{,}{v}\right]$
 ${R}{≔}\left[\left[{1}{,}{0}\right]{,}\left[{1}{,}{-}\frac{{1}}{{2}}\right]\right]$
 ${E}{≔}\left[{x}{=}{-5}{,}{y}{=}{3}\right]$ (1)
 > $a≔\mathrm{PuiseuxSeries}\left(p,X,U,R,E\right)$
 ${a}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{\left(\frac{{{x}}^{{2}}}{\sqrt{{y}}}{+}{1}\right){}{{y}}^{{3}}}{{{x}}^{{5}}}{:}\frac{{{y}}^{{3}}}{{{x}}^{{5}}}{+}\frac{{{y}}^{{5}}{{2}}}}{{{x}}^{{3}}}\right]$ (2)
 > $b≔\mathrm{Inverse}\left(a\right)$
 ${b}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{{x}}^{{5}}}{\left(\frac{{{x}}^{{2}}}{\sqrt{{y}}}{+}{1}\right){}{{y}}^{{3}}}{:}\frac{{{x}}^{{5}}}{{{y}}^{{3}}}{+}{\dots }\right]$ (3)
 > $c≔ab$
 ${c}{≔}\left[{PuisⅇuxSⅇriⅇs:}{1}\right]$ (4)
 > $\mathrm{Truncate}\left(c,15\right)$
 ${1}$ (5)

Note that truncating a Puiseux series truncates its inner power series: the terms are homogeneous in the variables $u,v$ of the inner power series, but not necessarily in the variables $x,y$ of the Puiseux series itself.

We can also compute the inverse $b$ by specifying the rational function that is the inverse of the polynomial $p$ and the appropriate E.

 > $\mathrm{mp}≔\left[u=x{y}^{0},v=x{y}^{-\frac{1}{2}}\right];$$E≔\left[x=5,y=-3\right]$
 ${\mathrm{mp}}{≔}\left[{u}{=}{x}{,}{v}{=}\frac{{x}}{\sqrt{{y}}}\right]$
 ${E}{≔}\left[{x}{=}{5}{,}{y}{=}{-3}\right]$ (6)
 > $b≔\mathrm{PuiseuxSeries}\left(\frac{1}{1+uv},\mathrm{mp},E\right)$
 ${b}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{{x}}^{{5}}}{\left(\frac{{{x}}^{{2}}}{\sqrt{{y}}}{+}{1}\right){}{{y}}^{{3}}}{:}\frac{{{x}}^{{5}}}{{{y}}^{{3}}}{+}{\dots }\right]$ (7)
 > $c≔ab$
 ${c}{≔}\left[{PuisⅇuxSⅇriⅇs:}{1}\right]$ (8)
 > $\mathrm{Truncate}\left(c,15\right)$
 ${1}$ (9)

Create a Puiseux series with the expression ${ⅇ}^{x}$ as internal power series.

 > $g≔\mathrm{PowerSeries}\left(d↦\frac{{\left(u+v\right)}^{d}}{d!},\mathrm{analytic}=\mathrm{exp}\left(u+v\right)\right)$
 ${g}{≔}\left[{PowⅇrSⅇriⅇs of}{{ⅇ}}^{{u}{+}{v}}{:}{1}{+}{\dots }\right]$ (10)
 > $\mathrm{mp}≔\left[u={x}^{\frac{1}{4}},v={x}^{\frac{3}{5}}{y}^{-\frac{2}{5}}\right]$
 ${\mathrm{mp}}{≔}\left[{u}{=}{{x}}^{{1}}{{4}}}{,}{v}{=}\frac{{{x}}^{{3}}{{5}}}}{{{y}}^{{2}}{{5}}}}\right]$ (11)
 > $b≔\mathrm{PuiseuxSeries}\left(g,\mathrm{mp}\right)$
 ${b}{≔}\left[{PuisⅇuxSⅇriⅇs of}{{ⅇ}}^{{{x}}^{{1}}{{4}}}{+}\frac{{{x}}^{{3}}{{5}}}}{{{y}}^{{2}}{{5}}}}}{:}{1}{+}{\dots }\right]$ (12)
 > $\mathrm{Truncate}\left(b,5\right)$
 ${1}{+}{{x}}^{{1}}{{4}}}{+}\frac{{{x}}^{{3}}{{5}}}}{{{y}}^{{2}}{{5}}}}{+}\frac{\sqrt{{x}}}{{2}}{+}\frac{{{x}}^{{17}}{{20}}}}{{{y}}^{{2}}{{5}}}}{+}\frac{{{x}}^{{6}}{{5}}}}{{2}{}{{y}}^{{4}}{{5}}}}{+}\frac{{{x}}^{{3}}{{4}}}}{{6}}{+}\frac{{{x}}^{{11}}{{10}}}}{{2}{}{{y}}^{{2}}{{5}}}}{+}\frac{{{x}}^{{29}}{{20}}}}{{2}{}{{y}}^{{4}}{{5}}}}{+}\frac{{{x}}^{{9}}{{5}}}}{{6}{}{{y}}^{{6}}{{5}}}}{+}\frac{{x}}{{24}}{+}\frac{{{x}}^{{27}}{{20}}}}{{6}{}{{y}}^{{2}}{{5}}}}{+}\frac{{{x}}^{{17}}{{10}}}}{{4}{}{{y}}^{{4}}{{5}}}}{+}\frac{{{x}}^{{41}}{{20}}}}{{6}{}{{y}}^{{6}}{{5}}}}{+}\frac{{{x}}^{{12}}{{5}}}}{{24}{}{{y}}^{{8}}{{5}}}}{+}\frac{{{x}}^{{5}}{{4}}}}{{120}}{+}\frac{{{x}}^{{8}}{{5}}}}{{24}{}{{y}}^{{2}}{{5}}}}{+}\frac{{{x}}^{{39}}{{20}}}}{{12}{}{{y}}^{{4}}{{5}}}}{+}\frac{{{x}}^{{23}}{{10}}}}{{12}{}{{y}}^{{6}}{{5}}}}{+}\frac{{{x}}^{{53}}{{20}}}}{{24}{}{{y}}^{{8}}{{5}}}}{+}\frac{{{x}}^{{3}}}{{120}{}{{y}}^{{2}}}$ (13)

If any of the vectors in R or any of the exponent vectors in mp are not grevlex greater than zero, an error is signaled.

 > $R≔\left[\left[1,0\right],\left[-1,1\right]\right]$
 ${R}{≔}\left[\left[{1}{,}{0}\right]{,}\left[{-1}{,}{1}\right]\right]$ (14)
 > $\mathrm{PuiseuxSeries}\left(p,X,U,R,E\right)$
 > $\mathrm{mp}≔\left[u={x}^{\frac{1}{4}},v={x}^{-5}{y}^{-5}\right]$
 ${\mathrm{mp}}{≔}\left[{u}{=}{{x}}^{{1}}{{4}}}{,}{v}{=}\frac{{1}}{{{x}}^{{5}}{}{{y}}^{{5}}}\right]$ (15)
 > $\mathrm{PuiseuxSeries}\left(g,\mathrm{mp}\right)$

References

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

Compatibility

 • The MultivariatePowerSeries[PuiseuxSeries] command was introduced in Maple 2023.