# Computational solutions to families of systems of linear equations

## Question

Does there exist a computer package that will solve families of systems of linear equations over a field of prime characteristic?

## An Example

Suppose I wanted to know when the following system of equations had a non-zero solution x over a field of characteristic p:

$\binom{N+i}{i} x =0 \quad \forall \quad 1 \leq i \leq M$

where $N,M \geq 1$. Obviously if I choose $N,M,p$ there are any number of packages that will work. But is there anything that will tell me in general when there is a solution, that is: There is a solution if and only if $p^{\ell(M)} \mid N+1$, where $\ell(M)$ is minimal such that $p^{\ell(M)} >M$.

## Motivation

Over a field of characteristic $p$, it is easy to show that the Specht module $S^{(n-a,a)}$ has a submodule $D^{(n)}$ if and only if $\binom{n-a+1}{1},\ldots,\binom{n}{a} \equiv 0 \mod p$; this reduces to the problem above. One can also look at $Hom(S^\lambda,S^\mu)$. In some cases, this can also be reduced to solving systems of linear equations which are easy (for a computer) to solve for individual problems but it seems hard to find a general condition on $\lambda, \mu$. (Subject to constraints on the partitions - the absolutely general case is very far away.) Even where explicit solutions for the family of equations can be found, the proof is usually by case-by-case analysis, hence my desire for a computer program to do the work.

If you think a package will work, could you give me some indication of how to implement it, for example for the family of systems above. Hopefully I can then adapt that to solve the problem I actually want!

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To clarify: I don't want a solution to the example, the solution is given at the end of the example. What I am after is a computer package that will solve similar but more complicated systems. If I could find a package that solves the example, I can use it as a starting point. – Sinead Lyle Nov 3 '10 at 9:53

I am not sure if I am interpreting the question correctly, but if all you want is to decide whether some binomial coefficients are zero modulo $p$, then you shouldn't need a software package. You can use the following old elementary result, originally due to Lucas. If $n = \sum a_i p^i, m = \sum b_i p^i, 0 \le a_i, b_i < p$ are integers given by their base $p$-expansion, then ${n \choose m} \equiv \prod {a_i \choose b_i} \mod p$. In particular, ${n \choose m} \equiv 0 \mod p$ if and only if $a_i < b_i$ for some $i$.

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I'm afraid that I don't want a solution to the example: actually, the solution is given in the example. What I want is a computer package that will solve similar but more complicated systems. – Sinead Lyle Nov 3 '10 at 9:47

"LinBox provides tools for linear algebra computations over the integers, the rational numbers, and finite fields and rings."

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But here the characteristic, for example, is one of the unknowns :) – Mariano Suárez-Alvarez Nov 1 '10 at 19:47
The problem is that I am looking to solve families of equations. In the example, I know packages which will give me solutions for any N,M,p that I feed in. But I do not know any way of solving general systems of equations. If you know how to do this in LinBox, could you let me know? – Sinead Lyle Nov 3 '10 at 9:50

If I interpret the question correctly, the determinant you are looking for is computed exactly in http://arxiv.org/abs/math/9902004, Section 2.

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I'm afraid that I don't want a solution to the example: actually, the solution is given in the example. What I want is a computer package that will solve similar but more complicated systems. – Sinead Lyle Nov 3 '10 at 9:51