Modifiers

The following functions modify the representation of the power series or apply a simple automorphism.

ScaleGenerators(s, ls): RngPowAlgElt, SeqEnum -> RngPowAlgElt

Let \(\{\gamma_i\}_i\) be the basis (determined from the representation chosen by Magma) of the exponent lattice of the series \(s\), and let \(\sigma: {\underline x}^{\gamma_i} \mapsto ls[i] {\underline x}^{\gamma_i}\). Return the series \(\sigma(s)\).

ChangeRing(s, R): RngPowAlgElt, RngMPol -> RngPowAlgElt

If \(R\) is a multivariate polynomial domain compatible with the approximation domain Domain(s), return the same power series with new approximation domain \(R\). This is sort of a “coercion between power series rings”.

SimplifyRep(s): RngPowAlgElt -> RngPowAlgElt

Factorizing: BoolElt                    Default: true

“Simplifies” the internal representation of a series. The result will be a series of atomic type without recursive (substitution) dependencies on other power series. The defining polynomial of the simplified series will be irreducible and therefore a minimal polynomial over Domain(s) (unless Factorizing is false when it will only be guaranteed to be squarefree). After the simplification, DefiningPolynomial returns this polynomial, which can be useful (e.g., for IsPolynomial). However, experience shows that the resulting representation is in general neither simple nor more efficient for subsequent computations.

There is a dangerous pitfall:

Assume we have a series represented by a tree with nodes of type A and B. Assume further that the leaves have been constructed by RationalPuiseux with parameter Gamma set to some value. Then the intention was probably to work over the subring of a polynomial ring with restricted support. If now SimplifyRep, with Factorizing as true, is called, then a minimal polynomial over the whole polynomial ground ring is computed which is maybe not what one wants.

Example: scale (ex-e60c95)

We can modify s2 by mapping generators (of Laurent polynomials) \(x^{1/5}y^{-2/5} \mapsto 3 x^{1/5}y^{-2/5}\) and \(x^{2/5}y^{1/5} \mapsto 4 x^{2/5}y^{1/5}\).

> Expand(ScaleGenerators(s2, [3,4]), 15);
true
64/81*x^2*y^11 - 64/3*x^5*y^5 - 16/9*x^2*y^6 + 48*x^5 + 4*x^2*y

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One can naturally view h1 as a series in \({\mathbb{Q}}(i)[[u,v]]\).

> Qi<i> := NumberField(R.1^2 + 1) where R is PolynomialRing(Q);
> Qiuv<u,v> := PolynomialRing(Qi, 2, "glex");
> h4 := ChangeRing(s1, Qiuv);
> Expand(h4, 4); Domain(h4);
true u^3 + 3*u^2*v + 3*u*v^2 + v^3 + u^2 + 2*u*v + v^2 + u + v
Polynomial ring of rank 2 over Qi
Graded Lexicographical Order
Variables: u, v

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We have seen that the power series h3 is zero, but its representation does not show this immediately. We can “explicitize” its representation.

> SimplifyRep(h3 : Factorizing := true);
Algebraic power series
0
> DefiningPolynomial($1);
z

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