Constructors

Using constructors one can construct power series starting from polynomial data or using other power series recursively.

PolyToSeries(s): RngMPolElt -> RngPowAlgElt

Given a multivariate polynomial \(s\), returns the series representation of \(s\).

AlgebraicPowerSeries(dp, ip, L, e): RngUPolElt, RngMPolElt, Lat, RngIntElt -> RngPowAlgElt

AlgebraicPowerSeries(dp, ip, e): RngUPolElt, RngMPolElt, RngIntElt -> RngPowAlgElt

AlgebraicPowerSeries(dp, ip): RngUPolElt, RngMPolElt -> RngPowAlgElt

subs: SeqEnum                    Default: []

Define a power series root of a polynomial \(p\) using an initial expansion \(ip\) and its exponent lattice \(1/\hbox{\tt e}~\hbox{\tt L}\). The defining polynomial \(p\) is either \(dp\) when \(subs\) is empty or obtained by substituting the elements of \(subs\) into the variables of \(dp\). The initial expansion has to be sufficiently long in order to uniquely identify a root, see [Beck, 2007, Cond. 4.3]. In this initial finite expansion and in subsequent ones, the variables occurring actually represent \(e\)-th roots, ie \(x_1x_2^2\) is really \(x_1^{1/e}x_2^{2/e}\). All exponents of monomials occurring in these expansions and the coefficients of \(p\) should lie in \(L\) (although this is not checked in many places), monomials of \(p\) giving true values rather than \(e\)-th roots.

There are simpler constructors where \(L\) is omitted (when it is assumed to be the standard integral lattice) or \(e\) is (when it is taken as 1).

As noted earlier, no strong checks are performed at construction time on the correctness of \(L\) or \(e\) or whether \(ip\) is indeed the initial expansion of a unique root in this “raw data” constructor. Incorrect initial data will only be revealed when failure occurs in further expansion of the series. The preferred methods of series creation are ImplicitFunction, EvaluationPowerSeries and RationalPuiseux because in these cases the sanity checks are more easily verified.

EvaluationPowerSeries(s, nu, v): Tup, SeqEnum, SeqEnum -> RngPowAlgElt

Given a series \(s\), a sequence \(nu\) of vectors in the dual of its exponent lattice of \(s\) and a sequence \(v\) (of the same length) of power series in some other common domain (with compatible coefficient field). Returns the series obtained by substituting \({\underline x}^{\mu} \mapsto \prod_i \hbox{\tt v[i]}^{\langle \hbox{\tt nu[i]}, \mu \rangle}\). This requires that \(nu\) and \(v\) fulfill a certain condition on the orders to guarantee convergence of the resulting series, see [Beck, 2007, Cond. 4.6].

ImplicitFunction(dp): RngUPolElt -> RngPowAlgElt

subs: SeqEnum                    Default: []

The unique series with zero constant term defined by a polynomial \(p \in k[x_1,..,x_n][z]\) or \(k[[x_1,\dots,x_n]][z]\), fulfilling the conditions of the implicit function theorem, i.e., \(p(0,\dots,0)=0\) and \(\partial p / \partial z(0,\dots,0) \neq 0\). The polynomial \(p\) is equal to \(dp\) possibly substituted with the series in \(subs\) as in AlgebraicPowerSeries. \(dp\) should have coefficients in a multivariate polynomial ring.

Rational Puiseux Expansions

Let \(p \in k[[x_1,\dots,x_n]][z]\) be a quasi-ordinary polynomial over a field \(k\) of characteristic zero. This means that \(p\) is non-zero, squarefree and monic (i.e., its leading coefficient in \(z\) is a unit in the power series ring) and if \(d \in k[[x_1,\dots,x_n]]\) denotes its discriminant then \(d = x_1^{e_1} \cdots x_n^{e_n} u(x_1,\dots,x_n)\) where \(u\) is a unit in the power series ring.

In this case the Theorem of Jung-Abhyankar states that \(p\) has \(\deg(p)\) distinct Puiseux series roots, i.e., power series roots with fractionary exponents and coefficients in the algebraic closure of \(k\).

These roots are computed by a generalization of the so called Newton-Puiseux algorithm. Also Duval’s extension for computing rational parametrization has been implemented.

RationalPuiseux(p): RngUPolElt -> Tup, SeqEnum, RngIntElt

Gamma       : Lattice                      Default: StandardLattice
subs        : SeqEnum                      Default: []
Duval       : BoolElt                      Default: false
OnlySingular: BoolElt                      Default: false
ExtName     : MonStgElt                    Default: "gamma"
ExtCount    : RngIntElt                    Default: 0

We first specify the behavior of this function in the case that no special value of \(subs\) has been given. This function assumes that \(p\) is a univariate polynomial over a multivariate polynomial ring \(S = k[x_1, ..., x_r]\) and that \(p\) is quasi-ordinary. In this case it will compute a set of rational parametrizations of \(p\). Note that for reasons of efficiency the user has to make sure that \(p\) is actually quasi-ordinary! (Otherwise, further processing of the output may result in runtime errors.)

The first return value will be the exponent lattice of the input polynomial in the usual format \(<\Gamma_0, e_0>\). If the parameter Gamma has been specified, then \(\Gamma_0 = \hbox{\tt Gamma}\) and \(e_0 = 1\). In this case Gamma has to be an integral ‘r’-dimensional lattice of full rank containing all the exponents of \(p\). Otherwise \(\Gamma_0\) will be set to the \(r\)-dimensional standard lattice and again \(e_0=1\).

As a second value a complete list of rational parametrizations in the format \(<\lambda, s, N, E>\) is returned. Here \(\lambda\) is a sequence of \(r\) field elements and \(s\) is a fractionary algebraic power series of type RngPowAlgElt. Let \(p_1\) denote the image of \(p\) under the transformation \(x^{\mu_i} \mapsto \lambda_i x^{\mu_i}\) where \((\mu_i)_i\) is the basis of the exponent lattice \(e_0^{-1} \Gamma_0\) then \(s\) is a solution of \(p_1\), i.e., we have \(p_1(s) = 0\). Note that if neither Gamma nor subs have been supplied this just means that \(x_i\) is substituted by \(\lambda_i x_i\). Finally \(N\) is the index of \(e_0^{-1} \Gamma_0\) in the exponent lattice of \(s\) and \(E\) is the degree of the extension of the coefficient field needed for defining \(s\).

The behavior described above corresponds to the Newton-Puiseux algorithm with Duval’s trick. The field extensions that are used for expressing the series fulfill a certain minimality condition. If Duval is set to false then the function returns a complete set of representatives (up to conjugacy) of Puiseux series roots of the original polynomial \(p\), in other words, the \(\lambda\)-vectors will always be vectors of ones.

If OnlySingular is set to true then only those parametrizations that correspond to singular branches are returned.

If the ground field has to be extended, the algebraic elements will be assigned the name \(\hbox{\tt ExtName}\_i\) where \(i\) starts from ExtCount. The last return value is the value of ExtCount plus the number of field extensions that have been introduced during the computation.

Finally, if the parameter subs is passed, then it has to be a sequence of \(r\) power series in a common domain and internally the variables in \(p\) will be substituted by the corresponding series. Again the resulting polynomial has to be quasi-ordinary. In this case \(\Gamma_0\) and \(e_0\) are determined by building the sum of the exponent lattices of all series in subs. The parameter Gamma then has no effect.

For further details on the algorithm and other references see [Beck, 2007, Sec. 4.3]

Example: constructors (ex-da4682)

We illustrate the constructors by examples. For displaying results we already use the command Expand that will be explained later.

> Q := Rationals(); Qs<s> := FunctionField(Q);
> Qxy<x,y> := PolynomialRing(Q, 2, "glex");
> Qxyz<z> := PolynomialRing(Qxy);
> Qst<t> := PolynomialRing(Qs, 1, "glex");
> Qstu<u> := PolynomialRing(Qst);

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One can consider polynomials as series.

> s0 := PolyToSeries(1 - 3*x + x^2*y + y^20);
> Expand(s0, 10);
true x^2*y - 3*x + 1

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One can define series by the implicit function theorem at the origin.

> s1 := ImplicitFunction(z*(1 - x - y) - x - y);
> Expand(s1, 4);
true x^3 + 3*x^2*y + 3*x*y^2 + y^3 + x^2 + 2*x*y + y^2 + x + y

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One can define a power series if an initial expansion is known. Note that the following power series has exponent lattice \({\mathbb{Z}}({{1}\over{5}},-{{2}\over{5}}) + {\mathbb{Z}}({{2}\over{5}},{{1}\over{5}})\) but its “expansions” are polynomials supported on \({\mathbb{Z}}(1,-2) + {\mathbb{Z}}(2,1)\).

> defpol := (1+5*y+10*y^3+10*y^2+5*y^4+y^5)*z^5+(-20*y^3*x-
>   30*y^2*x-5*y^4*x-5*x-20*y*x)*z^4+(10*x^2+30*y^2*x^2+10*y^3*x^2+
>   30*x^2*y)*z^3+(-20*y*x^3-10*x^3-10*y^2*x^3)*z^2+
>   (5*y*x^4+5*x^4)*z-x^5-x^2*y;
> Gamma := Lattice(RMatrixSpace(Integers(), 2, 2) ! [1,-2, 2,1]);
> init := x^2*y;
> s2 := AlgebraicPowerSeries(defpol, init, Gamma, 5);
> Expand(s2, 20);
true
-x^2*y^16 + x^5*y^10 + x^2*y^11 - x^5*y^5 - x^2*y^6 + x^5 + x^2*y

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We can “substitute” series into each other.

> X := AlgebraicPowerSeries(u^3-t+s*t^2, t, StandardLattice(1), 3);
> Y := PolyToSeries(t);
> duals := [RSpace(Integers(), 2) | [1, 3], [2, 1]];
> s3 := EvaluationPowerSeries(s2, duals, [X, Y]);
> Expand(s3, 13);
true (-1/9*s^2 - 1/3*s - 1)*t^10 + (-1/3*s + 1)*t^7 + t^4

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We can compute all the Puiseux series roots of a quasi-ordinary polynomial up to conjugacy over \({\mathbb{Q}}\).

> qopol := z^6 + 3*x*y^2*z^4 + x*y*z^3 + 3*x^2*y^4*z^2 + x^3*y^6;
> _, prms := RationalPuiseux(qopol : Duval := false); prms;
[*
    <[ 1, 1 ], Algebraic power series -x*y, 3, 1>,
    <[ 1, 1 ], Algebraic power series gamma_0*x*y, 3, 2>,
    <[ 1, 1 ], Algebraic power series -x^2*y^5, 3, 1>,
    <[ 1, 1 ], Algebraic power series gamma_1*x^2*y^5, 3, 2>
*]
> Domain(prms[2][2]); ExponentLattice(prms[2][2]);
Polynomial ring of rank 2 over Number Field with defining
polynomial $.1^2 - $.1 + 1 over the Rational Field
Graded Lexicographical Order
Variables: x, y
<
    Lattice of rank 2 and degree 2
    Basis:
    ( 1  1)
    ( 1 -2),

    3
>
> Expand(prms[2][2], 15);
true x^3*y^9 + (gamma_0 - 1)*x^2*y^5 + gamma_0*x*y

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We find that the sum over all field extensions \(1+2+1+2 = 6\) is equal to the degree of the defining polynomial qopol. The third parametrization involves a field extension of \({\mathbb{Q}}\) by gamma_0 s.t. \(\hbox{\tt gamma\_0}^2 - \hbox{\tt gamma\_0} + 1 = 0\) and an extension of the exponent lattice to \({\mathbb{Z}}({{1}\over{3}},{{1}\over{3}}) + {\mathbb{Z}}({{1}\over{3}},-{{2}\over{3}})\). It turns out that the field extension is not necessary if we are only interested in parametrizations.

> _, prms := RationalPuiseux(qopol : Duval := true); prms;
[*
    <[ -1, -1 ], Algebraic power series -x*y, 3, 1>,
    <[ -1, 1 ], Algebraic power series -x^2*y^5, 3, 1>
*]

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No field extensions have been introduced, but this required the application of automorphisms \({\mathbb{Q}}[[x, y]]\to {\mathbb{Q}}[[x, y]]\) in advance (more precisely \(x \mapsto -x, y \mapsto -y\) resp. \(x \mapsto -x, y \mapsto y\)). This time we can sum up the overall extension degrees (i.e., for fields and lattices) \(3 \cdot 1 + 3 \cdot 1 = 6\) to the degree of qopol.