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 ${\mathrm{\mathbb{Q}}}^p$.

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 ${\mathrm{\mathbb{Q}}}^q$ 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 ${\mathrm{\mathbb{Q}}}^p$,

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 ${\mathrm{\mathbb{Q}}}^p$.

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.

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

$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]$

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

$X≔\left[x\,y\right]$

$U≔\left[u\,v\right]$

$R≔\left[\left[1\,0\right]\,\left[1\,-\frac{1}{2}\right]\right]$

$E≔\left[x=\mathrm{-5}\,y=3\right]$

$a≔\mathrm{PuiseuxSeries}\left(p\,X\,U\,R\,E\right)$

$a≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{\left(\frac{x^2}{\sqrt{y}}+1\right)y^3}{x^5}\mathrm{:}\frac{y^3}{x^5}+\frac{y^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{x^3}\right]$

$b≔\mathrm{Inverse}\left(a\right)$

$b≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{x^5}{\left(\frac{x^2}{\sqrt{y}}+1\right)y^3}\mathrm{:}\frac{x^5}{y^3}+\mathrm{\dots }\right]$

$c≔ab$

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

$\mathrm{Truncate}\left(c\,15\right)$

$1$

$\mathrm{mp}≔\left[u=x{y}^0\,v=x{y}^{-\frac{1}{2}}\right]\;$$E≔\left[x=5\,y=-3\right]$

$\mathrm{mp}≔\left[u=x\,v=\frac{x}{\sqrt{y}}\right]$

$E≔\left[x=5\,y=\mathrm{-3}\right]$

$b≔\mathrm{PuiseuxSeries}\left(\frac{1}{1+uv}\,\mathrm{mp}\,E\right)$

$b≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{x^5}{\left(\frac{x^2}{\sqrt{y}}+1\right)y^3}\mathrm{:}\frac{x^5}{y^3}+\mathrm{\dots }\right]$

$c≔ab$

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

$\mathrm{Truncate}\left(c\,15\right)$

$1$

$g≔\mathrm{PowerSeries}\left(d\mapsto \frac{{\left(u+v\right)}^d}{d!}\,\mathrm{analytic}=\exp \left(u+v\right)\right)$

$g≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^{u+v}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{mp}≔\left[u=x^{\frac{1}{4}}\,v=x^{\frac{3}{5}}{y}^{-\frac{2}{5}}\right]$

$\mathrm{mp}≔\left[u=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\,v=\frac{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}{y^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}\right]$

$b≔\mathrm{PuiseuxSeries}\left(g\,\mathrm{mp}\right)$

$b≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}{\ⅇ}^{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+\frac{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}{y^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(b\,5\right)$

$1+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+\frac{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}{y^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{\sqrt{x}}{2}+\frac{x^{\raisebox{1ex}{$17$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}}{y^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}{2y^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{6}+\frac{x^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}}{2y^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$29$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}}{2y^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}{6y^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x}{24}+\frac{x^{\raisebox{1ex}{$27$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}}{6y^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$17$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}}{4y^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$41$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}}{6y^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$12$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}{24y^{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{120}+\frac{x^{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}{24y^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$39$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}}{12y^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$23$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}}{12y^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^{\raisebox{1ex}{$53$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}}{24y^{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}+\frac{x^3}{120y^2}$

$R≔\left[\left[1\,0\right]\,\left[-1\,1\right]\right]$

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

$\mathrm{PuiseuxSeries}\left(p\,X\,U\,R\,E\right)$

Error, (in MultivariatePowerSeries:-PuiseuxSeries) all the rays in [[1, 0], [-1, 1]] must be grevlex([x, y]) positive

$\mathrm{mp}≔\left[u=x^{\frac{1}{4}}\,v=x^{-5}{y}^{-5}\right]$

$\mathrm{mp}≔\left[u=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\,v=\frac{1}{x^5{y}^5}\right]$

$\mathrm{PuiseuxSeries}\left(g\,\mathrm{mp}\right)$

Error, (in MultivariatePowerSeries:-PuiseuxSeries) all the rays in [[1/4, 0], [-5, -5]] must be grevlex([x, y]) positive

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

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

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