Hi Tim, I just opened #10771 for that purpose. Algorithm is proposed below.
On 11 Feb., 10:55, daly <d...@axiom-developer.org> wrote: > On Fri, 2011-02-11 at 01:49 -0800, Simon King wrote: > > Hi, > > > On 11 Feb., 09:56, Simon King <simon.k...@uni-jena.de> wrote: > > Does anyone have a better idea? Would it be a correct definition if > > one insisted on reduced fractions? > > > That's why I was asking for an algorithm for gcd and lcm > in the subdomain. What do you mean by "subdomain"? Do you mean, an integral domain R as a subdomain of Frac(R)? > The unit (1) is correct but not by your definition, and > apparently not helpful for the original poster. Any unit is a correct answer from the point of view of a PID. That's why it makes sense to use the freedom. Here is an algorithm: --------- ASSUMPTIONS: R is an integral domain, F is its fraction field. gcd and lcm are defined for R INPUT: x,y: Elements of F. Since there is gcd in R, we can assume that x.numerator() and y.denominator() are relatively prime, i.e., x=a/b and y=c/d for a,b,c,d in R, the two fractions being reduced. OUTPUT: gcd(x,y) returns gcd(x.numerator(),y.numerator())/ lcm(x.denominator(),y.denominator()) lcm(x,y) returns lcm(x.numerator(),y.numerator())/ gcd(x.denominator(),y.denominator()) ----------------- PROPERTIES: - Up to units in R, we have gcd(x,y)*lcm(x,y) = gcd(a,c)/ lcm(b,d)*lcm(a,c)/gcd(b,d) = a*c/b*d = x*y - Moreover, again up to units in R, we have gcd(a/1,b/1) = gcd(a,b)/ lcm(1,1) = gcd(a,b) and lcm(a/1,b/1) = lcm(a,b)/gcd(1,1) = lcm(a,b) My questions to people that get more sleep than I: Does that algorithm makes sense, mathematically? At least it seems to me that the gcd-definition is exactly what is used in Maxima: sage: P.<x> = QQ[] sage: a = (1+x^2)*(1+2*x)^2 sage: b = 1-x^4 sage: c = (1+3*x^2)*(1+2*x) sage: d = 1-x^5 sage: x = a/b sage: y = c/d sage: factor(gcd(maxima(x),maxima(y))) (2*x+1)/((x-1)*(x+1)*(x^4+x^3+x^2+x+1)) sage: factor(gcd(x.numerator(),y.numerator())/ lcm(x.denominator(),y.denominator())) (x - 1)^-1 * (x + 1)^-1 * (x + 1/2) * (x^4 + x^3 + x^2 + x + 1)^-1 Best regards, Simon -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
