Peter-Lawrence . Montgomery on Fri, 26 Feb 1999 01:17:22 +0100 (MET)

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Re: Strange polresultant behavior

Roland Dreier <> reports

> I'm getting the following strange behavior from gp version 2.0.13.  I have
> two polynomials in two variables, x and y (over the field with 5
> elements).  I take their resultant with respect to x to get a big
> polynomial in y.  I then take a root of this polynomial and plug it in for
> y in the original two polynomials to get two polynomials in x.  Unless I'm
> confused, the resultant of these two new polynomials should be zero -- it
> doesn't matter whether I plug the root in after taking the resultant or
> before.  But for certain polynomials, I don't get zero.
        I didn't try Roland's example, but the problem may be a double
projective root in x, if the two leading coefficints vanish for the same
value of y.  When we substitute a numerical value for y before
taking the resultant, the degrees change.

                    GP/PARI CALCULATOR Version 2.0.11 (beta)
                             unknown 32-bit version
                (readline disabled, extended help not available)

                           Copyright (C) 1989-1998 by
          C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier.

Send bug reports, suggestions and patches to
Type ? for help.

   realprecision = 38 significant digits
   seriesprecision = 16 significant terms
   format = g0.38

parisize = 10000000, primelimit = 500000, buffersize = 30000
   echo = 1 (on)
? f1=x*y-1
%1 = y*x - 1
? f2=x*y-2
%2 = y*x - 2
? polresultant(f1,f2,x)
%3 = -y
? quit
Good bye!

        By the way, why are blanks removed in echo mode?  
My input to this example was

f1 = x*y - 1
f2 = x*y - 2
polresultant(f1, f2, x)