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! >> >> >> >> -- 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/710cf4ab-c782-4ad0-a195-fcd47ed1abc5%40googlegroups.com.
