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Hello Derivers,

Concerning the polynomial p0:=x^8-40*x^6+352*x^4-960*x^2+576
it indeed satisfies the very unpleasant relation
EXPAND(FACTOR(p0,Complex,x)) /= p0

I found a very curious fact in Derive 6 with this polynomial.
(In fact I use for the moment a beta version).
The minimal polynomial in Z[X] having a root SQRT(a)+SQRT(b)+SQRT(c)
can be obtained as:

p3(a,b,c,x):=FIRST(GROEBNER_BASIS([u^2-a,v^2-b,w^2-c,x-u-v-w],[u,v,w,x]))

Our polynomial p0 is p3(2,3,5,x).

Defining also:

testzero(p,x):=EXPAND(FACTOR(p,Complex,x))-p

shouldbe0(a,b,c,x):=testzero(p3(a,b,c,x))

then shouldbe0(2,3,5,x) is simplified to a nonzero polynomial.
But for several other integer values I have tested
(including shouldbe0(1377,1112,573,x)), shouldbe0(a,b,c,x) simplifies
to 0.
My question is: what is so special with p0?
(or maybe with 2-3-5). Does some philosophical problem hide here?

Cheers,
Valeriu



> I get the polinomial correctly
> 
> x^8 - 40x^6 + 352x^4 - 960x^2 + 576 (#1)
> 
> by squaring from three times, isolating th roots.
> 
> x = sqrt(2) + sqrt(3) + sqrt(5)
> 
> But when I factorize #1 in radicals with Derive, it put:
> 
> (x + <(20·<6 + 4·<114))·(x - <(20·<6 + 4·<114))·(x + <(20·<6 - 4·<114))·(x -
> <(20·<6 - 4·<114))·(x^2 + x·<(40·<6 - 48) + 24)·(x^2 - x·<(40·<6 - 48) + 24)
> = 0
> 
> ('<' is 'sqrt')
> 
> And in complex,
> 
> (x + <(20·<6 + 4·<114))·(x - <(20·<6 + 4·<114))·(x + <(20·<6 - 4·<114))·(x -
> <(20·<6 - 4·<114))·(x + <(10·<6 - 12) + î·<(36 - 10·<6))·(x + <(10·<6 -
> 12) - î·<(36 - 10·<6))·(x - <(10·<6 - 12) + î·<(36 - 10·<6))·(x - <(10·<6 -
> 12) - î·<(36 - 10·<6)) = 0
> 
> getting four real roots, any of them sqrt(2) + sqrt(3) + sqrt(5), and four
> complex conjugates ones.
> 
> But substituing x ---> sqrt(t), we get
> 
> t^4 - 40·t^3 + 352·t^2 - 960·t + 576
> 
> that derive factorice correctly to:
> 
> (t + 2·<15 + 2·<10 - 2·<6 - 10)·(t + 2·<15 - 2·<10 + 2·<6 - 10)·(t - 2·<15 +
> 2·<10 + 2·<6 - 10)·(t - 2·<15 - 2·<10 - 2·<6 - 10)
> 
> or, aproximating
> 
> (t - 0.8284574727)·(t - 3.679609142)·(t - 6.522431886)·(t - 28.96950149)
> 
> getting four linear real factors, as It expects.
> 
> My version of Derive is 5.06 in Spanish.
> 
> Saludos,
> 
> Ignacio Larrosa Caņestro
> A Coruņa (Espaņa)
> [log in to unmask]
> 




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