8.6 Groebner basis computation with rational function coefficients

Such variables that appear within the input polynomials but not appearing in the input variable list are automatically treated as elements in the coefficient field by top level functions, such as gr().

[64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
[(-e*a+d*b)*x-f*b+e*c,(-e*a+d*b)*y+f*a-d*c]

In this example, variables a, b, c, and d are treated as elements in the coefficient field. In this case, a Groebner basis is computed on a bi-variate polynomial ring F[x,y] over rational function field F = Q(a,b,c,d). Notice that coefficients are considered as a member in a field. As a consequence, polynomial factors common to the coefficients are removed so that the result, in general, is different from the result that would be obtained when the problem is considered as a computation of Groebner basis over a polynomial ring with rational function coefficients. And note that coefficients of a distributed polynomial are limited to numbers and polynomials because of efficiency.

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