I have no idea whether SAGE supports this, but basically what you need is 
fraction-free Gaussian elimination: See section 3.2.3 of 
http:/staff.bath.ac.uk/masjhd/JHD-CA.pdf

On Wednesday, 22 August 2012 13:04:08 UTC+1, Erik Aas wrote:
>
> I'm trying to solve a system of linear equations over the field 
> Q(x1,...,xn) of rational functions in x1,...,xn.
>
> The system is Ax = 0 where A has entries from the field and has rather 
> large size (500 rows and columns for example).
>
> One way to do it is to compute kernel(A). In my case it is 1-dimensional. 
> The basis vector returned has coordinates of the form 
> (polynomial/polynomial + polynomial) / (polynomial - polynomial) et.c. 
> Calling .denominator() and .numerator() on the expression does not 
> terminate before my patience does. Is there a way to make sure the equation 
> solver keep all coefficients on the form polynomial / polynomial?
>
>

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