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. 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! > > > > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/edce4769-1ca3-4400-bd50-ef01f0a11d5c%40googlegroups.com.
