def is_hyperbolic(self, p):
r"""
Check if the quadratic form is a sum of hyperbolic planes over
the `p`-adic numbers `\QQ_p` or over the real numbers `\RR`.
REFERENCES:
This criteria follows from Cassels's "Rational Quadratic Forms":
- local invariants for hyperbolic plane (Lemma 2.4, p58)
- direct sum formulas (Lemma 2.3, p58)
INPUT:
- `p` -- a prime number > 0 or `-1` for the infinite place
OUTPUT:
boolean
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1])
sage: Q.is_hyperbolic(-1)
False
sage: Q.is_hyperbolic(2)
False
sage: Q.is_hyperbolic(3)
False
sage: Q.is_hyperbolic(5) ## Here -1 is a square, so it's true.
True
sage: Q.is_hyperbolic(7)
False
sage: Q.is_hyperbolic(13) ## Here -1 is a square, so it's true.
True
"""
## False for odd-dim'l forms
if self.dim() % 2:
return False
## True for the zero form
if not self.dim():
return True
## Compare local invariants
## Note: since the dimension is even, the extra powers of 2 in
## self.det() := Det(2*Q) don't affect the answer!
m = ZZ(self.dim() // 2)
if p == -1:
return self.signature() == 0
if p == 2:
return (QQ(self.det() * (-1)**m).is_padic_square(p)
and self.hasse_invariant(p) == (-1)**m.binomial(2)
) # here -1 is hilbert_symbol(-1,-1,2)
return (QQ(self.det() * (-1)**m).is_padic_square(p)
and self.hasse_invariant(p) == 1)
def is_hyperbolic(self, p):
r"""
Check if the quadratic form is a sum of hyperbolic planes over
the `p`-adic numbers `\QQ_p` or over the real numbers `\RR`.
REFERENCES:
This criteria follows from Cassels's "Rational Quadratic Forms":
- local invariants for hyperbolic plane (Lemma 2.4, p58)
- direct sum formulas (Lemma 2.3, p58)
INPUT:
- `p` -- a prime number > 0 or `-1` for the infinite place
OUTPUT:
boolean
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1])
sage: Q.is_hyperbolic(-1)
False
sage: Q.is_hyperbolic(2)
False
sage: Q.is_hyperbolic(3)
False
sage: Q.is_hyperbolic(5) ## Here -1 is a square, so it's true.
True
sage: Q.is_hyperbolic(7)
False
sage: Q.is_hyperbolic(13) ## Here -1 is a square, so it's true.
True
"""
## False for odd-dim'l forms
if self.dim() % 2:
return False
## True for the zero form
if not self.dim():
return True
## Compare local invariants
## Note: since the dimension is even, the extra powers of 2 in
## self.det() := Det(2*Q) don't affect the answer!
m = ZZ(self.dim() // 2)
if p == -1:
return self.signature() == 0
if p == 2:
return (QQ(self.det() * (-1) ** m).is_padic_square(p) and
self.hasse_invariant(p) ==
(-1) ** m.binomial(2)) # here -1 is hilbert_symbol(-1,-1,2)
return (QQ(self.det() * (-1) ** m).is_padic_square(p) and
self.hasse_invariant(p) == 1)
