From: &noreply@axelvogt.de   
      
   On 05.04.2014 23:42, clicliclic@freenet.de wrote:   
   ...   
   >>> Maybe you can do this faster than Derive 6.10. I want to eliminate the   
   >>> two variables x1, x2 from the three polynomial equations   
   ...   
   >>> Thus I started GROEBNER_BASIS([lhs1, lhs2, lhs3], [x1, x2, x3, x4]) on   
   >>> Derive, where lhs1, lhs2, lhs3 are the left-hand sides of the three   
   >>> equations; the computation is still running. However, I am interested   
   >>> only in the basis elements not involving x1 and x2.   
   >>   
   ...   
   > I want to know in particular if the bivariate elements for the   
   > higher-order problems possess a similar structure.   
      
   Let be J the ideal defined by lhs1, lhs2, lhs3 and genJ its 3 generators.   
      
   Maple 18 says this defines a curve (dim = 1 over the complex numbers),   
   but I have not had the patients to wait for possible components.   
      
      
   It suggests ordering x3, x1, x4, x2 and finds a basis w.r.t. that,   
   denote it by groebnerJ, it is a list of 10 (!) polynomials. Then:   
      
   map(length, groebnerJ); # ~ 10^7 characters overall   
      
      [23,1826,21386,19882,175151,172745,177725,175800,177278,176374]   
      
   The first entry is lhs3 (of course (?)).   
      
      
   The degrees in x_i for the base generators are as follows:   
      
      map('q -> degree(q,x1)', groebnerJ);   
                        [0, 9, 8, 8, 8, 8, 8, 8, 8, 8]   
      
      map('q -> degree(q,x2)', groebnerJ);   
                   [2, 18, 72, 72, 71, 71, 71, 71, 71, 70]   
      
      map('q -> degree(q,x3)', groebnerJ);   
                        [0, 0, 1, 1, 1, 1, 1, 2, 2, 3]   
      
      map('q -> degree(q,x4)', groebnerJ);   
                        [2, 0, 1, 1, 1, 1, 1, 1, 1, 1]   
      
      
   I have not checked for the bivariate terms (so far).   
      
   PS: send me a PM if you want the base as zipped text file.   
      
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