Actually $0 is the currently executing program, which varies system to system. See perlvar for more detail.
On Solaris 8, it's likely to be something like /usr/bin/perl /path/to/perl/script, not at all like the filename given by warn or die.
On Linux (or at least this Linux box) it is the script name.
Update: Correction; after re-reading perlvar myself, it states "$0...Contains the name of the file containing the Perl script being executed".
However, my comment about it varying OS to OS stands.
Update 2: I am an idiot. I've double checked, following vek's post (++); he's quite right. $0 does exactly as advertised.
All I need to do now is remember where I saw a script reported as I originally and incorrectly reported - ps probably. *Sigh*
If the information in this post is inaccurate, or just plain wrong, don't just downvote - please post explaining what's wrong.
That way everyone learns.