create a Puiseux series
PuiseuxSeries(g, X, U, R, e)
PuiseuxSeries(g, mp, e)
polynomial, rational function, or power series generated by this package
(optional) list of ordered variables for the Puiseux series
(optional) list of ordered variables for the power series
(optional) list of grevlex positive rational rays
(optional) list of equations representing the exponents of a monomial multiplying the Puiseux series
list of equations representing the change of variables to be applied to g
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≔x1,…,xp and U≔u1,…,um be ordered lists of variables.
Let R≔r1,…,rm be a list of m grevlex-positive p-dimensional rational vectors.
Let e≔e1,…,ep be a point in ℚp.
Let g⁡U≔∑n=0∞gn⁡U be a multivariate power series in U with homogeneous components gn⁡U.
For any v=v1,…,vq in ℚq and any list Y=y1,…,yq, we write Yv for y1v1⁢…⁢yqvq. Moreover, we write XR for the list Xr1,…,Xrm of m products of powers of the variables in X. Then P≔Xe⁢g⁡XR is a Puiseux series, and every Puiseux series can be written in this way. This can be understood as evaluating g⁡U at ui=Xri and then multiplying the result by Xe.
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 ℂC⟦X⟧,
R is a list of grevlex positive p-dimensional rays contained in ℚp,
E is a list of the form x1=e1,…,xp=ep with e=e1,…,ep in ℚp.
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.
Create a Puiseux series, determine its inverse, multiply them and find its truncation to homogeneous degree 15.
p ≔ PowerSeries⁡1+u⁢v;X ≔ x,y;U ≔ u,v;R ≔ 1,0,1,−12;E ≔ x=−5,y=3
a ≔ PuiseuxSeries⁡p,X,U,R,E
a≔PuⅈsⅇuxSⅇrⅈⅇs of x2y+1⁢y3x5 : y3x5+y52x3
b ≔ Inverse⁡a
b≔PuⅈsⅇuxSⅇrⅈⅇs of x5x2y+1⁢y3 : x5y3+…
c ≔ a⁢b
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.
mp ≔ u=x⁢y0,v=x⁢y−12;E ≔ x=5,y=−3
b ≔ PuiseuxSeries⁡11+u⁢v,mp,E
Create a Puiseux series with the expression ⅇx as internal power series.
g ≔ PowerSeries⁡d→u+vdd!,analytic=ⅇu+v
g≔PowⅇrSⅇrⅈⅇs of ⅇu+v : 1+…
mp ≔ u=x14,v=x35⁢y−25
b ≔ PuiseuxSeries⁡g,mp
b≔PuⅈsⅇuxSⅇrⅈⅇs of ⅇx14+x35y25 : 1+…
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 ≔ 1,0,−1,1
Error, (in MultivariatePowerSeries:-PuiseuxSeries) all the rays in [[1, 0], [-1, 1]] must be grevlex([x, y]) positive
mp ≔ u=x14,v=x−5⁢y−5
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.
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