On Sun, Oct 20, 2019 at 11:38 AM Simon Brandhorst <sbrandho...@web.de> wrote:
>
> Dear Ignat Soroko,
>
> the quadratic forms code was written with quadratic forms over QQ and over ZZ
> in mind. So I would be very sceptic about any functionality over number
> fields.
> For instance the signature vector you mention does not make sense of the
> F.<a> =CyclotomicField(8). Instead of a single signature vector, over a
> number field you should
> obtain a signature for each real embedding F.
CyclotomicField(8) has no real embedding.
Cyclotomic fields come with a default embedding:
The default embedding sends the generator to the complex primitive
n^{th} root of unity of least argument.
sage: CC(k.gen())
0.623489801858734 + 0.781831482468030*I
which is not real (and in general cyclotomic fields have no real embedding)
However, having a real embedding seems to be an unnecessary restriction, as
a symmetric matrix of real cyclotomics will have real eigenvalues, and
it has a signature.
(I don't know whether the signature will stay invariant if the
embedding changes - it should be either
easy to prove or easy to give a counterexample...)
Thanks,
Dima
> Since F has no real places it does not really have signatures. All infinite
> places are complex and all regular quadratic forms over CC are equivalent. So
> basically
>
> sage:
> Q=QuadraticForm(K,8,[1/2,-a/2,0,0,0,0,0,0,1/2,-a/2,0,0,0,0,0,1/2,-1/2,0,0,
> ....: 0,0,1/2,-1/2,0,0,0,1/2,-1/2,0,0,1/2,-a/2,0,1/2,-a/2,1/2])
> sage: Q.signature_vector()
>
> should either raise a value error, or
> be called signature_vectors() and return a dictionary of real places and
> signature vectors.
>
> Best,
> Simon
>
> On Friday, October 18, 2019 at 2:12:02 AM UTC+2, Ignat Soroko wrote:
>>
>> I am computing the signature of a quadratic form having entries 0, 1, -1/2,
>> -sqrt(2)/2. I noticed that the result of signature_vector() is different if
>> we treat the number sqrt(2) as a cyclotomic or as a real number. Please look
>> at the example:
>>
>> sage: K.<z>=CyclotomicField(8)
>> sage: a=z-z^3 # a is a square root of 2
>> sage: a-sqrt(2)
>> 0
>> sage:
>> Q=QuadraticForm(K,8,[1/2,-a/2,0,0,0,0,0,0,1/2,-a/2,0,0,0,0,0,1/2,-1/2,0,0,
>> ....: 0,0,1/2,-1/2,0,0,0,1/2,-1/2,0,0,1/2,-a/2,0,1/2,-a/2,1/2])
>> sage: Q.signature_vector()
>> (8, 0, 0)
>>
>> this cannot be true since there exists an isotropic vector:
>>
>> sage: v=vector([1,a,1,0,0,0,0,0])
>> sage: v*Q.matrix()*v
>> 0
>>
>> Let's try it over reals:
>>
>> sage: a=sqrt(2)
>> sage:
>> Q=QuadraticForm(RR,8,[1/2,-a/2,0,0,0,0,0,0,1/2,-a/2,0,0,0,0,0,1/2,-1/2,0,0
>> ....: ,0,0,1/2,-1/2,0,0,0,1/2,-1/2,0,0,1/2,-a/2,0,1/2,-a/2,1/2])
>> sage: Q.signature_vector()
>> (6, 2, 0)
>>
>> however, the isotropic vector above is not isotropic anymore:
>>
>> sage: v=vector([1,a,1,0,0,0,0,0])
>> sage: v*Q.matrix()*v
>> sqrt(2)*(1.00000000000000*sqrt(2) - 1.41421356237310) -
>> 1.41421356237310*sqrt(2) + 2.00000000000000
>>
>> I also tried to define
>>
>> sage: a=sqrt(AA(2))
>> sage:
>> Q=QuadraticForm(AA,8,[1/2,-a/2,0,0,0,0,0,0,1/2,-a/2,0,0,0,0,0,1/2,-1/2,0,0
>> ....: ,0,0,1/2,-1/2,0,0,0,1/2,-1/2,0,0,1/2,-a/2,0,1/2,-a/2,1/2])
>>
>> but Q.signature_vector() gives a runtime error with many lines of code
>> ending in:
>>
>> RuntimeError: maximum recursion depth exceeded
>>
>>
>> Questions:
>> 1) is Q.signature_vector() over cyclotomic field is interpreted in some
>> other way than for reals, thus making the result (8,0,0) somehow correct?
>>
>> 2) Which setting would guarantee both the correct result for
>> signature_vector() using the exact arithmetic and at the same time show that
>> v is actually an isotropic vector?
>>
>> Thank you!
>>
>>
>>
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