push(@factored_rows,%coefficient_array);
[download]

should be

push(@factored_rows, \%coefficient_array);
[download]

otherwise a hash gets flattened in a list of keys and values. Without '\' it's as if you wrote if like that:

push(@factored_rows, 'hash_key1', $value1, 'hash_key2', $value2);
[download]

(hash is not an array, btw, they are totally different things)

Also, your program is hard to read. Some advice:

@factored_eqns = &factorize(\@eqn_set,\@variables);
[download]

'&' in factorize is completely unnecessary.

my @factored_eqns;
...
@factored_eqns = factorize(\@eqn_set,\@variables);
[download]

Don't declare a bunch of variables all at the top of the program/sub. They're not much better then global variables when used that way (and it's very write-only). Perl is not C89. Instead of:

my @factored_eqns;
...
@factored_eqns = factorize(\@eqn_set,\@variables);
...

    my $term;
    ...
    foreach $term(@{$equation}) {
    ...
[download]

do

my @factored_eqns = factorize(\@eqn_set,\@variables);
...

    foreach my $term (@{ $equation }) {
    ...
[download]

Indent your code properly. Rather then

    foreach $equation (@this_eqn_set){ 
    #then for each variable in the eqution
    foreach $variable (@this_variable_set){
    ...
[download]

Write it like so:

    for my $equation (@this_eqn_set) {
        for my $variable (@this_variable_set) {
            ...
[download]

(for and foreach are the same thing)

Uncomment strict. And add use warnings;

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