Bug in is_hyperbolic

[pkoprowski]

The method is_hyperbolic in the class of quadratic form returns incorrect results over the field QQ_2. Here is an explicit example:

q = DiagonalQuadraticForm(QQ, [1,1,-1,-1])
print q.is_hyperbolic(2)

The form <1,1,-1,-1> is clearly hyperbolic - by definition. It is a sum of two hyperbolic planes. Nevertheless Sage returns False here.

The reason is as follows. In the file quadratic_forms/quadratic_form__local_field_invariants.py in function is_hyperbolic the actual code is

    elif p == 2:
        return QQ(self.det() * (-1)**m).is_padic_square(p) and (self.hasse_invariant(p) == (-1)**m)    ## Actually, this -1 is the Hilbert symbol (-1,-1)_p

while it should be

    elif p == 2:
        return QQ(self.det() * (-1)**m).is_padic_square(p) and (self.hasse_invariant(p) == (-1)**(m*(m-1)/2))    ## Actually, this -1 is the Hilbert symbol (-1,-1)_p

For mathematical explanation see e.g. T.Y. Lam "Introduction to Quadratic Forms over Field", Proposition V.3.25

I'm not sure how to formally patch the code, so I'm posting it this way.

[MRupert]

This commit should fix the problem. I also added functionality and improved the documentation on the infinite place.

---

New commits:

​abb9f99

18430: Fixes self.is_hyperbolic(2) and various cleanups

[jdemeyer]

Can you replace

if p == "infinity":
    return self.is_definite()
else:
    ...

by

if p == Infinity:
    return self.is_definite()

...

This means you don't need to indent the whole block for p a prime number and I also prefer the actual value Infinity (which needs to be imported from sage.rings.infinity) instead of the string "infinity".

I also don't understand why you use -1 at one point and "infinity" somewhere else.

And instead of writing

(-1)**(m*(m-1)/2)) ## Actually, this -1 is the Hilbert symbol (-1,-1)

why don't you actually write

hilbert_symbol(-1, -1, p)**(m*(m-1)/2))

[chapoton]

rebase and clean-up

---

New commits:

​fc968e3

Merge branch 'u/MRupert/bug_in_is_hyperbolic' in 7.4.b2

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​64cfdf3	Merge branch 'u/chapoton/18430' in 8.0.b0
​7690f84	trac 18430 fixing doctests, plus little code cleanup

[tscrim]

Replying to jdemeyer:

I also don't understand why you use -1 at one point and "infinity" somewhere else.

And instead of writing

(-1)**(m*(m-1)/2)) ## Actually, this -1 is the Hilbert symbol (-1,-1)

why don't you actually write

hilbert_symbol(-1, -1, p)**(m*(m-1)/2))

I am pretty sure the former is (much) faster and the comment makes the simplification clear.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​3231a6d

trac 18430 details

[chapoton]

done

[chapoton]

ping ?

[tscrim]

If you just fix up this INPUT: docstring (in a few places):

-`p` -- a prime number > 0 or `-1` for the infinite place.
+- `p` -- a prime number > 0 or `-1` for the infinite place

Once that is done, you can set a positive review on my behalf.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b4f668a	Merge branch 'u/chapoton/18430' in 8.0.b9
​f99393f	trac 18430 INPUT field has no dot

[tscrim]

LGTM.