unfortunately, this doesn't really help: the main reason is that the roots
may not have an explicit expression in terms of radicals. See
https://trac.sagemath.org/ticket/32143
But even when explicit expressions exist, there seems to be a problem:
sage: var("y a")
(y, a)
sage: p = y^4 + y + a
sage: p.roots(y, multiplicities=False)[0]
-1/2*sqrt(1/3)*sqrt((4*a + 3*(1/6*sqrt(-256/3*a^3 + 9) +
1/2)^(2/3))/(1/6*sqrt(-256/3*a^3 + 9) + 1/2)^(1/3)) -
1/2*sqrt(-4/3*a/(1/6*sqrt(-256/3*a^3 + 9) + 1/2)^(1/3) -
(1/6*sqrt(-256/3*a^3 + 9) + 1/2)^(1/3) + 6*sqrt(1/3)/sqrt((4*a +
3*(1/6*sqrt(-256/3*a^3 + 9) + 1/2)^(2/3))/(1/6*sqrt(-256/3*a^3 + 9) +
1/2)^(1/3)))
sage: R = PolynomialRing(SR, "y")
sage: Rp.roots(multiplicities=False)
[]
emanuel.c...@gmail.com schrieb am Donnerstag, 8. Juli 2021 um 09:46:28
UTC+2:
> You may work on the univariate polynamial ring in your variable of
> interest over a suitable ring. A simple example :
>
> sage: var("x, y, z")
> (x, y, z)
> sage: foo=x^3-x*sin(y+z)+1
> sage: foo.polynomial(ring=PolynomialRing(SR,"x")).parent()
> Univariate Polynomial Ring in x over Symbolic Ring
> sage: foo.polynomial(ring=PolynomialRing(SR,"x")).roots()
> [(-1/6*(-I*sqrt(3) + 1)*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3) - 1/2*(I*sqrt(3) + 1)*(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3),
> 1),
> (-1/6*(I*sqrt(3) + 1)*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3) - 1/2*(-I*sqrt(3) + 1)*(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3),
> 1),
> (1/3*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) - 1/2)^(1/3) +
> (1/6*sqrt(-4/3*sin(y + z)^3 + 9) - 1/2)^(1/3),
> 1)]
>
> It might be interesting to create a more specific base ring, but only if
> your expression is indeed a polynomial (e. g. no function calls as in my
> oversimplified example…).
>
> It turns out that, in this precise case, it is not necessary to explicitly
> create a polynomial ; but doing so gives a two orders of magnitude speed
> gain (not exactly small potatoes) :
>
> sage: %time foo.roots()
> CPU times: user 3.83 s, sys: 40.1 ms, total: 3.87 s
> Wall time: 2.9 s
> [(-1/6*(-I*sqrt(3) + 1)*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3) - 1/2*(I*sqrt(3) + 1)*(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3),
> 1),
> (-1/6*(I*sqrt(3) + 1)*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3) - 1/2*(-I*sqrt(3) + 1)*(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3),
> 1),
> (1/3*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) - 1/2)^(1/3) +
> (1/6*sqrt(-4/3*sin(y + z)^3 + 9) - 1/2)^(1/3),
> 1)]
> sage: %time foo.polynomial(ring=PolynomialRing(SR,"x")).roots()
> CPU times: user 41.2 ms, sys: 17 µs, total: 41.2 ms
> Wall time: 41.1 ms
> [(-1/6*(-I*sqrt(3) + 1)*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3) - 1/2*(I*sqrt(3) + 1)*(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3),
> 1),
> (-1/6*(I*sqrt(3) + 1)*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3) - 1/2*(-I*sqrt(3) + 1)*(1/6*sqrt(-4/3*sin(y + z)^3 + 9) -
> 1/2)^(1/3),
> 1),
> (1/3*sin(y + z)/(1/6*sqrt(-4/3*sin(y + z)^3 + 9) - 1/2)^(1/3) +
> (1/6*sqrt(-4/3*sin(y + z)^3 + 9) - 1/2)^(1/3),
> 1)]
>
> HTH,
>
> Le lundi 5 juillet 2021 à 22:01:10 UTC+2, axio...@yahoo.de a écrit :
>
>> I am trying to fix #32133, which is about translating FriCAS expressions
>> containing symbolic roots of polynomials, given by their minimal
>> polynomials.
>>
>> This is easy enough if the polynomials are univariate, so the roots are
>> algebraic numbers and I can use the wonderful machinery of QQbar.
>>
>> However, I don't see what I can do with polynomials containing extra
>> variables.
>>
>> Any hints appreciated!
>>
>> Martin
>>
>
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