Re: [Axiom-math] Decomposition of rationnal fractions

[Nicolas FRANCOIS]

Le Fri, 14 May 2010 08:52:33 +0200,
Martin Rubey <address@hidden> a écrit :

> 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)))))

Yeah, definitely closer ! I'll have to investigate all this. Thanks.

\bye

-- 

Nicolas FRANCOIS                      |  /\ 
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