Accessors and Expansion

The following functions provide an interface to conveniently extract information from a power series defined as above.

Domain(s): RngPowAlgElt -> RngMPol

Return the multivariate polynomial ring that is used for approximating the series \(s\) by its truncations.

ExponentLattice(s): RngPowAlgElt -> Tup

Return the exponent lattice \((1/e)\Gamma\) of the series as tuple \((\Gamma, e)\) where \(\Gamma\) is an integral lattice and \(e\) is an integer.

DefiningPolynomial(s): RngPowAlgElt -> RngUPolElt

Return a defining polynomial of the series which is a squarefree univariate polynomial over the multivariate polynomial domain Domain(s). In the case of series defined with substitutions, the computation may be expensive and can involve recursive resultant computations.

Order(s): RngPowAlgElt -> RngIntElt

TestZero: BoolElt                    Default: false

Given a series \(s\), return the integral order (total degree of smallest non-zero term occurring) of its expansion as returned by Expand, i.e., its fractionary order times the exponent denominator. If \(s\) is zero, this function will not terminate. Set TestZero to true to get a return value \(-1\) in this case, but note that this involves the computationally complex call IsZero.

Expand(s, ord): RngPowAlgElt, RngIntElt -> BoolElt, RngMPolElt

Given the power series \(\beta\) which is represented by \(s\), let \(\alpha\) be the result of substituting variables \(x_i \mapsto x_i^e\) where \(e\) is taken from the output of ExponentLattice(s) (i.e., \(\alpha\) is \(\beta\) without exponent denominators). Returns true and the truncation of \(\alpha\) modulo terms of order greater or equal ord. A return of false indicates that the representation is inconsistent (which should only happen when RationalPuiseux is called with non quasi-ordinary input or AlgebraicPowerSeries is used inconsistently).

Example: accessors (ex-4231eb)

We can study the series s3.

> Domain(s3);
Polynomial ring of rank 1 over Univariate rational function
field over Rational Field
Graded Lexicographical Order
Variables: t
> ExponentLattice(s3);
<
    Standard Lattice of rank 1 and degree 1,

    3
>
> DefiningPolynomial(s3);
(s^45*t^45 - ... - 15*s^30*t^2 - s^30)*u^15 + ... +
(5*s^37*t^41 - ... + 120*s^31*t^19 - 30*s^30*t^18)*u^3 -
s^35*t^40 + ... - 5*s^31*t^21 + s^30*t^20
> Order(s3);
4

Run in calculatorCopy input

These commands reveal the following about s3: It is a power series in \({\mathbb{Q}}(s)[[t^{1/3} ]]\), because it is approximated in \({\mathbb{Q}}(s)[t]\) and has exponent lattice \({{1}\over{3}}{\mathbb{Z}}\). A defining polynomial in \({\mathbb{Q}}(s)[t][u]\) was also computed. (Recall that s3 has been defined recursively.) The order is \({{4}\over{3}}\) which we know already from a previous expansion.