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Re: [Axiom-math] Decomposition of rationnal fractions
From: |
Martin Rubey |
Subject: |
Re: [Axiom-math] Decomposition of rationnal fractions |
Date: |
Fri, 14 May 2010 08:52:33 +0200 |
User-agent: |
Gnus/5.11 (Gnus v5.11) Emacs/22.3 (gnu/linux) |
Nicolas FRANCOIS <address@hidden> writes:
> Hi.
>
> Is there any way to obtain the decomposition in simple elements (don't
> know exactly how to say this in english) of a fraction of the form :
>
> 1
> -------------------
> (1-X)(1-X^2)(1-X^5)
>
> (to obtain its formal series equivalent \sum a_nX^n, a_n being the
> number of ways to pay n€ using 1, 2 and 5€ corners (no, there
> is no such thing as a 5€ corner, but there's a 5€ banknote !)).
>
> I'd like to obtain the C-decomposition, what do I have to do ?
>
> More precisely : is there a way to force the use of an extension of
> Q(X), by adding roots like exp(2*I*PI/5) or sqrt(2) ?
>
> \bye
>
> PS : clearly I'm not very good at using Axiom documentation !
Is the following close to what you have in mind? (two problems: you
need to know the extension in advance, and I don't see a way to factor
over extensions of degree higher than one right now. Possibly Waldek
knows.)
(1) -> SAEs5 := SAE(FRAC INT,UP(s5,FRAC INT),s5^2-5)
(1)
SimpleAlgebraicExtension(Fraction(Integer),UnivariatePolynomial(s5,Fra
ction(Integer)),s5^2+-5)
Type: Type
(2) -> p:UP(x,SAEs5) :=(x^5-1)*(x^2-1)*(x-1)
8 7 6 5 3 2
(2) x - x - x + x - x + x + x - 1
Type:
UnivariatePolynomial(x,SimpleAlgebraicExtension(Fraction(Integer),UnivariatePolynomial(s5,Fraction(Integer)),s5^2+-5))
(3) -> factor p
3 2 1 1 2 1 1
(3) (x - 1) (x + 1)(x + (- - s5 + -)x + 1)(x + (- s5 + -)x + 1)
2 2 2 2
Type:
Factored(UnivariatePolynomial(x,SimpleAlgebraicExtension(Fraction(Integer),UnivariatePolynomial(s5,Fraction(Integer)),s5^2+-5)))
(4) -> partialFraction(1/p, x)
(4)
13 2 9 27 1 1 1 1 1
-- x - -- x + -- - (- -- s5 - --)x + -- s5 - --
40 10 40 8 50 10 50 10
----------------- - ----- + ----------------------------
3 x + 1 2 1 1
(x - 1) x + (- - s5 + -)x + 1
2 2
+
1 1 1 1
(-- s5 - --)x - -- s5 - --
50 10 50 10
--------------------------
2 1 1
x + (- s5 + -)x + 1
2 2
Type:
PartialFraction(UnivariatePolynomial(x,Fraction(Polynomial(SimpleAlgebraicExtension(Fraction(Integer),UnivariatePolynomial(s5,Fraction(Integer)),s5^2+-5)))))
Apart from that: Gröbner bases are in Axiom (FriCAS, OpenAxiom), and a
rather good implementation, too.
Martin