Thank you for figuring that out! Shall we alert the developers about it? It
is my first time on that forum, I am not sure I know how to do that...
And what about the behavior that occurs when we try to use the field AA:
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])
sage: Q.signature_vector()
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call last)
<ipython-input-146-fedf8894b12c> in <module>()
----> 1 Q.signature_vector()
/opt/sagemath-8.6/local/lib/python2.7/site-packages/sage/quadratic_forms/quadratic_form__local_field_invariants.pyc
in signature_vector(self)
329
330 """
--> 331 diag = self.rational_diagonal_form()
332 p = 0
333 n = 0
/opt/sagemath-8.6/local/lib/python2.7/site-packages/sage/quadratic_forms/quadratic_form__local_field_invariants.pyc
in rational_diagonal_form(self, return_matrix)
175 # Quadratic forms do not support immutability, so we need to
make
176 # a copy to be safe.
--> 177 Q = deepcopy(Q)
178
179 if return_matrix:
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in deepcopy(x, memo, _nil)
188 raise Error(
189 "un(deep)copyable object of type
%s" % cls)
--> 190 y = _reconstruct(x, rv, 1, memo)
191
192 memo[d] = y
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in _reconstruct(x, info,
deep, memo)
332 if state is not None:
333 if deep:
--> 334 state = deepcopy(state, memo)
335 if hasattr(y, '__setstate__'):
336 y.__setstate__(state)
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in deepcopy(x, memo, _nil)
161 copier = _deepcopy_dispatch.get(cls)
162 if copier:
--> 163 y = copier(x, memo)
164 else:
165 try:
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in _deepcopy_dict(x, memo)
255 memo[id(x)] = y
256 for key, value in x.iteritems():
--> 257 y[deepcopy(key, memo)] = deepcopy(value, memo)
258 return y
259 d[dict] = _deepcopy_dict
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in deepcopy(x, memo, _nil)
161 copier = _deepcopy_dispatch.get(cls)
162 if copier:
--> 163 y = copier(x, memo)
164 else:
165 try:
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in _deepcopy_list(x, memo)
228 memo[id(x)] = y
229 for a in x:
--> 230 y.append(deepcopy(a, memo))
231 return y
232 d[list] = _deepcopy_list
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in deepcopy(x, memo, _nil)
188 raise Error(
189 "un(deep)copyable object of type
%s" % cls)
--> 190 y = _reconstruct(x, rv, 1, memo)
191
192 memo[d] = y
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in _reconstruct(x, info,
deep, memo)
326 dictiter = None
327 if deep:
--> 328 args = deepcopy(args, memo)
329 y = callable(*args)
330 memo[id(x)] = y
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in deepcopy(x, memo, _nil)
161 copier = _deepcopy_dispatch.get(cls)
162 if copier:
--> 163 y = copier(x, memo)
164 else:
165 try:
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in _deepcopy_tuple(x, memo)
235 y = []
236 for a in x:
--> 237 y.append(deepcopy(a, memo))
238 d = id(x)
239 try:
... last 4 frames repeated, from the frame below ...
/opt/sagemath-8.6/local/lib/python2.7/copy.pyc in deepcopy(x, memo, _nil)
188 raise Error(
189 "un(deep)copyable object of type
%s" % cls)
--> 190 y = _reconstruct(x, rv, 1, memo)
191
192 memo[d] = y
RuntimeError: maximum recursion depth exceeded
sage:
On Friday, October 18, 2019 at 6:38:50 AM UTC-5, Frédéric Chapoton wrote:
>
> The bug over the cyclotomic field is caused by
>
>
> sage: K = CyclotomicField(8)
> sage: K(-1/16) > 0
> True
>
>
>
> Le vendredi 18 octobre 2019 13:24:51 UTC+2, Frédéric Chapoton a écrit :
>>
>> This works (in sage 8.9) when using
>>
>> K.<z>=QuadraticField(2)
>>
>>
>>
>> Le vendredi 18 octobre 2019 02:12:02 UTC+2, Ignat Soroko a écrit :
>>>
>>> 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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