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In Julia, there is Nemo.jl for high performance exact and symbolic linear algebra. At least that is the goal; it doesn't have the best algorithms yet in all cases, but it has some of them (and more are being developed). You can see some examples here: http://nemocas.org/benchmarks.html

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I really like that library for other reasons, but I couldn't find documentation for anything symbolic. Could you point to the right places? Feel free to follow up on other forums since this is getting pretty off topic.

You have to work with multivariate polynomials (possibly over number fields). If you have symbolic expressions in another format, you would have to manually convert them to that representation first, possibly after introducing extra variables for non-algebraic constants (which may or may not be a good idea depending on the specific circumstances). Having that kind of rewriting done automatically would certainly be a good thing!

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