Note that this introduces 0's through a fencepost error,
but they don't change which polynomial is represented. I
think this is fair, but if you think that is cheating, you
can not save that character:

sub p{
@m=1;for$p(@_){my@t;for$i(0..$#m){my$j;$t[$i+$j++]+=$_*$m[$i]for@$p}@m
+=@t}[@m]
}

The trick lies in finding ways to not work through explicit
lookups by index, and in finding ways to not access
arrays through references. In fact there is not a single
lookup by index of an element in an array reference. (It
was cheaper to create and manually increment the index
variable.)

BTW note that the statement of the rules anticipated and
forbade saving a character by ending with \@m without
making @m a private variable.

Finally at a request from chatter, here is the solution
broken out and commented:

sub p{
@m=1; # Start the product at 1.
for$p(@_){ # Loop over the polynomials.
my@t; # Create a private temp array.
for$i(0..@m){ # Loop over the indexes of @m.
my$j; # Create the *other* index var.
$t[$i+$j++]+= # Manually increment $j while..
# Adding to the index of @t..
$_*$m[$i] # 2 terms multiplied together..
for@$p # for all the terms in the..
} # other polynomial.
@m=@t # Make the temp array our new
} # product.
[@m] # Return our answer in the
} # desired form.