* Georgi Guninski <[email protected]> [2012-09-19 07:34:46 +0300]:
> 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. I don't think the article talks about the multivariate resultant. Imho the line you quote is just there to mention that there are generalizations of resultants to the multivariate case (it says "Alternatively", "More generally" close to that line). > 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 Afaik this computes the resultant of p1 and p2 as polynomials in QQ(x2,x3,x4)[x1]. As p1 and p2 are constant, they have no common root. > On the same example pari/gp returns 0: I don't know anything about GP, but it seems that it doesn't like to compute resultants when the variable is not in any of the polynomials (or do any of the variables have a special meaning?) ? p1=(x2)*(x3-x4);p2=x2*(x3-2*x4); ? polresultant(p1,p2,x1) %2 = 0 ? p1=(y)*(z-t);p2=y*(z-2*t); ? polresultant(p1,p2,x1) %4 = 1 julian > On Tue, Sep 18, 2012 at 05:22:59PM +0200, Julian Rüth wrote: > > Hi, > > > > I'm not sure if I understand what is counterintuitive about the results. > > > > * Georgi Guninski <[email protected]> [2012-09-18 16:55:37 +0300]: > > > sage: K.<x1,x2,x3>=PolynomialRing(QQ) > > > sage: p1=(x2-1)*(x3+2) > > > sage: p2=(x2-1)*(x3+3) > > > sage: p1.resultant(p2) > > > 1 > > This is the resultant of p1 and p2 w.r.t. x1 (the first variable of K). > > > > > sage: K_.<x2,x3>=PolynomialRing(QQ) > > > sage: p1_=K_(p1) > > > sage: p2_=K_(p2) > > > sage: p1_.resultant(p2_) > > > 0 > > The resultant of p1 and p2 w.r.t. x2 (the first variable of K_). > > > > > sage: gp.polresultant(gp(p1),gp(p2)) > > > 0 > > I'm not entirely sure what gp.polresultant() does, but it seems it > > computes the resultant w.r.t. variable() > > > > sage: gp(p1).variable() > > x2 > > > > The following is strange though: > > > > sage: gp.polresultant(p1,p2,x1) > > 0 # this should be 1? > > sage: gp.polresultant(p1,p2,x2) > > 0 > > sage: gp.polresultant(p1,p2,x3) > > x2^2 - 2*x2 + 1 > > > > Am I missing something here? > > > > julian > > > > -- > > 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. > > -- > 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. > -- 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.
