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/ccbac5a6-2ea8-4684-897d-297c083ae17c%40googlegroups.com.
