On Oct 18, 7:37 am, Urs Hackstein <[email protected]> wrote: > Let f be an element of the quotient field of the polynomial ring (in > one variable) over the complex numbers. > Our goal is to find representatives a and b in the polynomial ring > over the complex numbers such that f=a/b. Can sage do this for me? > (Simple commands like simplify or expand don' t work.)
If you know bounds on the degrees of a,b it becomes a simple interpolation/linear algebra problem. If a,b are guaranteed to exist of degree <= d then just choose n=2*d+2 evaluation points z_1,z_2,...,z_n and consider the equations a(z_i)-f(z_i)*b(z_i) = 0 (for i=1,...,n) That gives you n equations that are linear homogeneous in the coefficients of a,b. Just solve for those [you can write down a matrix representation of that system with about as much effort as introducing sufficient variables and letting sage figure out what system to solve]. You get a high degree of confidence that your candidate function is correct when you ensure your system is overdetermined (i.e., pick n larger). If you use floats you may have to settle for a least squares solution and verify that the match is very good. Once you have a candidate function, you can try and see if sage can simplify b(z)*f(z)-a(z) to 0. However, if you're working with floats that is virtually guaranteed to not work or produce garbage. -- To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
