62.3 Comparison of Rational Functions

  • f = g

    Two rational functions f and g are equal if the product Numerator(f )*Denominator(g ) equals Numerator(g )*Denominator(f ). 

    gap> x:=Indeterminate(Rationals,"x");;y:=Indeterminate(Rationals,"y");;
gap> f:=3+x*y+x^5;;g:=5+x^2*y+x*y^2;;
gap> a:=g/f;;
gap> b:=(g*f)/(f^2);;
gap> a=b;
true

  • f < g

    The ordering of rational functions is defined in several steps. Monomials (products of indeterminates) are sorted first by total-degree and then lexicographically (see MonomialTotalDegreeLess). Products of monomials with ring elements (``terms'') are compared first by their monomials and then by their coefficients. 

    gap> x<y; 
true
gap> x^2*y<x*y^2;                  
true
gap> x*y<x^2*y;
true
gap> x^2*y < 5* y*x^2;
true
    Polynomials are compared by comparing the largest terms in turn until they differ. 
    gap> x+y<y;
false
gap> x<x+1;
true
    Rational functions are compared by comparing the polynomial Numerator(f )*Denominator(g ) with the polynomial Numerator(g )*Denominator(f ). (As the ordering of monomials used by GAP is invariant under multiplication this is independent of common factors in numerator and denominator.) 
    gap> f/g<g/f;
false
gap> f/g<(g*g)/(f*g);
false

    For univariate polynomials this reduces to an ordering first by total degree and then lexicographically on the coefficients.

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    GAP 4 manual
    February 2000