Arithmetic#

There are functions to perform basic arithmetic operations (addition, subtraction etc.) on power series.

AlgComb(c, ss): RngMPolElt, SeqEnum -> RngPowAlgElt#: Given a polynomial \(c\) in \(r\) variables and a sequence \(ss\) of \(r\) power series (in a common domain with compatible coefficient field) return the series obtained by substituting the elements of \(ss\) for the variables of \(c\). This allows the construction of completely arbitrary algebraic combinations.

s + t: RngPowAlgElt, RngPowAlgElt -> RngPowAlgElt#
s - t: RngPowAlgElt, RngPowAlgElt -> RngPowAlgElt#
s * t: RngPowAlgElt, RngPowAlgElt -> RngPowAlgElt#: Add, subtract or multiply two power series.

Example: arith (ex-46bd4b)#

One can easily substitute power series into polynomials.

> // construct the series s0^2+s1^2
> h0 := AlgComb(x^2 + y^2, [s0,s1]);
> Expand(h0, 3);
true 10*x^2 + 2*x*y + y^2 - 6*x + 1

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This includes of course the ring operations.

> h1 := Add(s1, PolyToSeries(One(Qxy)));
> Expand(h1, 4);
true
x^3 + 3*x^2*y + 3*x*y^2 + y^3 + x^2 + 2*x*y + y^2 + x + y + 1
> h2 := Mult(h1, PolyToSeries(1 - x - y));
> Expand(h2, 4);
true 1
> h3 := Add(h2, PolyToSeries(-One(Qxy)));
> Expand(h3, 4);
true 0

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