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How to make a machine accessible from the LAN using its hostname

Here are details of the machine I want to access using its hostname:

$ hostname
$ cat /etc/hosts   localhost   hostname.company.local  hostname

It's a default Debian 6 (Squeeze) install, so I didn't fiddle with anything yet.

This is what I get from a machine (running Debian Unstable) trying to access above machine:

$ ping hostname
ping: unknown host hostname
$ ping hostname.company.local
ping: unknown host hostname.company.local
$ cat /etc/resolv.conf
search company.local