On Thursday, 20 September 2012 19:05:56 UTC+8, Georgi Guninski wrote: > > pari disagrees with sage and maxima agrees with it. > > which way is it? > > maxima session: > (%i12) p1:(x2)*(x3-x4);p2:x2*(x3-2*x4); > (%i14) resultant(p1,p2,x1); > (%o14) 1 > (%i15) resultant(p1,p2,x2); > (%o15) 0 > > On Wed, Sep 19, 2012 at 07:34:46AM +0300, Georgi Guninski wrote: > > Hi, > > > > I may be missing something, but the resultant = 1 confuses me. >
I bet the confusion comes from the fact the Sylvester matrix in this case is empty. Whether an empty matrix has determinant 0, or 1, it's a deep question :–) Note that in the ring of empty matrices, 0 equals 1, so in this sense it's OK... sage: K.<x1,x2,x3,x4>=QQ[] sage: p1,p2=(x2)*(x3-x4),x2*(x3-2*x4) sage: p1.sylvester_matrix(p2,x1) [] sage: p1.sylvester_matrix(p2,x1).det() 1 > > According to wikipedia [1] > > the multivariate resultant or Macaulay's resultant of n homogeneous > polynomials in n variables is a polynomial in their coefficients that > vanishes when they have a common non-zero solution > > My pain is $1$ can't vanish while solutions exist. > > > > Here is homogeneous example: > > sage: K.<x1,x2,x3,x4>=QQ[] > > sage: p1,p2=(x2)*(x3-x4),x2*(x3-2*x4) > > sage: p1.resultant(p2,x1) > > 1 > > > > Certainly p1 and p2 have common solutions while the res. w.r.t. > > x1 never vanishes (got this in a real world situation). > > > > On the same example pari/gp returns 0: > > ? p1=(x2)*(x3-x4);p2=x2*(x3-2*x4);polresultant(p1,p2,x1) > > %5 = 0 > > > > > > [1]: > > http://en.wikipedia.org/w/index.php?title=Resultant&oldid=511538674 > > > > > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-support?hl=en.
