On 20 March 2018 at 23:57, Dima Pasechnik <[email protected]> wrote:
>
>
> On Tuesday, March 20, 2018 at 4:06:46 PM UTC, John Cremona wrote:
>>
>> Working with your degree 8 polynomial over Q is almost certainly better.
>> I would also recommend reducing the defining polynomial first:
>>
>> sage: R.<g> = QQ[]
>> sage: pol = g^8 - 5661818/709635*g^7 + 11951452814641/503581833225*g^6 -
>> 5464287298588/167860611075*g^5 + 42311165180509/503581833225*g^4 +
>> 290446480816/167860611075*g^3 + 6817133713732/503581833225*g^2 -
>> 11294971392/55953537025*g + 2238425344/503581833225
>> sage: K.<a> = NumberField(pol)
>> sage: K1=K.optimized_representation()[0]; K1
>> Number Field in a1 with defining polynomial x^8 - 2*x^7 -
>> 2073127276349*x^6 - 585042438455127612*x^5 + 17251120619520968221641540*x^4
>> + 47323235466058260399591984538122*x^3 +
>> 52569579991119152255555179191805210311*x^2
>> + 26979907667586120684167115024265757878264932*x +
>> 5304889912416030130201287805372669997413025784321
>>
>> -- not obviously a lot better, but at least it has integer coefficients.
>> We can easily find its Galois group abstractly:
>>
>
> Thanks---I was not aware about optimized_representation().
>
Underneath it's the pari function "polred" which does the work. It's a
good way of getting a nicer polynomial defining the same number field. In
most cases, calling it with two isomorphic fields will return the same nice
version; not always, so there is the version polredabs which is guaranteed
to always give the same output for any polynomial defining the field. (We
use this in the LMFDB.)
>
> Let me explain a bit what I'm trying to do.
>
> Given 7 points in P^2 with coefficients in Q, I need
> 1) to find an irreducible cubic q through them s.t. q intersects qbar
> (it's complex conjugate) in 9 distinct real points, and
>
Assuming that your 7 points are in general position (no three on a line, no
6 on a conic), the cubics through them form a 2-dimensional family (they
are all linear combinations of 3 of them, up to scaling). One way to get
your q would be to take two such rational cubics q1 and q2 which intersect
in 9 points and then let q=q1+i*q2, which intersects qbar in the same 9
points.
> 2) to get hold of the points of the intersection of q and qbar with the
> conic c={x^2+y^2+z^2=0}, and construct
> cubics r and rbar through these 12 points so that q*qbar+r*rbar is
> divisible by x^2+y^2+z^2; (and r should not vanish on any of the
>
9 points of intersection of q and qbar...)
> I am told this is called a complex structure on the conic c, but OK; I
> need it to do some real algebraic geometry. :-)
>
> I've done 1) by choosing 2 points on c with coordinates in Q[i], and they
> together with the original 7 points
> give me q, and their conjugates give me qbar, and they have the real
> intersection, as I need.
>
That sounds better than my construction in that I would expect the
intersection s of q and c not to lie in Q(i).
>
> Now, for 2) I have the "ugly degree 4 polynomial" over Q[i] mentioned
> above,
> that specifies the intersection of q and c, (I know 2 intersection
> points, and this gives me the remaining 4). The similar situation is for
> qbar.
> Now, I can "go symbolic", i.e. solve these degree 4 equations in radicals,
> and use these radicals for computation.
> Or I could construct an appropriate field extension, where these roots
> live...
>
> Perhaps there is a better way to do this, I don't know.
>
>
I have in the past successfully used QQbar to find intersection points of
two curves each defined over Q. Once you have an intersection point with
QQbar coordinates you can find out what number field it is defined over and
then go back to find the points again, working over that field. I used to
do that a lot with Magma's AlgebraicallyClosedFIeld and it also works using
QQbar.
John
>
>
>>
>> sage: K1.galois_group(type='pari')
>> Galois group PARI group [1152, -1, 47, "[S(4)^2]2"] of degree 8 of the
>> Number Field in a1 with defining polynomial x^8 - 2*x^7 - 2073127276349*x^6
>> - 585042438455127612*x^5 + 17251120619520968221641540*x^4 +
>> 47323235466058260399591984538122*x^3 +
>> 52569579991119152255555179191805210311*x^2
>> + 26979907667586120684167115024265757878264932*x +
>> 5304889912416030130201287805372669997413025784321
>>
>> That code means its a double cover of S4^2 (1152 = 2*24^2). That's
>> bigger than the 32 you were expecting though, so perhaps working with
>> relative extensions would be better after all.
>>
>> John
>>
>>
>> On 20 March 2018 at 15:44, Dima Pasechnik <[email protected]> wrote:
>>
>>> I do not know how efficiently relative extensions of Q[i] are
>>> implemented.
>>> I would not mind an absolute field.
>>> I need things like computing resultants and factoring univariate
>>> polynomials to work not too slowly.
>>>
>>> I did some Groebner basis computation that gave me what I suspect is an
>>> superfield of what I need, and it is of degree 32 over Q, with really huge
>>> coefs....
>>>
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>>
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