# Efficient derivation of null space of large symbolic matrices?

Hi all,

I'm wondering if anyone is aware of an efficient mechanism by which to derive the null space of a "large" symbolic matrix. Here, large means on the order of 10^2 rows, not necessarily square, where element (i,j) is a polynomial in symbolic parameters p_1,...,p_n, n also on the order of 10^2.

Conventional desktop symbol toolboxes like Matlab, Maxima, and Mathematica tend to break down at this problem size. Any pointers most welcome! Thanks.

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## 1 Answer

Try to specialize some of the variables. Given enough specializations, one can then reconstruct kernel elements by interpolation.

Another method: try to use series expansions in terms of the variables and lift the order of such series expansions.

Another useful trick (when working with rational coefficients or coefficients in a number field) is working over finite fields since large determinant computations use up huge amounts of memory. Use then either the Hensel lemma (working over $p-$adics) or combine the information coming from different primes in order to lift the solution to $\mathbb Z$ or $\mathbb Q$.

Last ressort: If everything fails and if you are really interested in just one very specific example, write a C-program for just your example using the best strategy you know and hope that it works.

There is probably no universally optimal way to do this, I guess you have to make advantage of any special features of your example.

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