On Wednesday, January 4, 2017 at 6:25:02 PM UTC+2, [email protected] wrote:
>
> I have two algebraic numbers defined by minimal polynomials:
>
> x, z = symbols('x, z')
> p = Poly((x-1)*(x-2)*(x-3)*(x-4))
> q = Poly((x-5)*(x-6)*(x-7)*(x-8))
>
> and I would like to compute the sum of these numbers. I [found](
> http://math.stackexchange.com/a/155132/327863) that I need to "`z=x+y` is
> a root of the resultant of `P(x)` and `Q(z−x)` (where we take this
> resultant by regarding `Q` as a polynomial in only `x`)".
>
> I'm totally new to all this algebraic numbers thing so I don't quite
> understand the advice but I tried:
>
> resultant(p, q.subs(x, z-x))
>
The resultant is used to eliminate one variable which has to be given as an
argument. So this is what you want:
resultant(p, q.subs(x, z - x), x)
>
> but then I got stuck. Please, could someone explain to me:
>
> - I would like to see the steps that lead to the computation of desired
> minimal polynomial with the help of resultant. I think I defined the two
> numbers properly but I don't know how to express that `Q(z−x)`.
>
> - As it stands now, the `resultant` function returns bivariate polynomial
> but I would've assumed that the resulting minimal polynomial should be
> univariate. How do I get rid of the second variable?
>
> - The above link also says "`P(x) = Q(y) = 0`". Does it mean that `p` and
> `q` can't be both `Poly((x-1)*(x-2)*(x-3)*(x-4))`? What if I would like to
> add the same two numbers?
>
> Thank you very much in advance!
>
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