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().
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
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.
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.
>
> 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] <javascript:>>
> 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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