"GDM cannot write to your authorization file...etc"
There are several posts which ask this question, so I'll post my own problem and solution and hope a search will help someone!
I meant to delete all of the files in my /tmp directory but actually ended up deleting the whole directory.
When I rebooted, I got the message "GDM cannot write to your authorization file...etc"
I knew by using the df command that my partitions were not full.
Some other posts mentioned something about permissions.
That got me thinking: perhaps my user login was trying to write to /tmp but that directory had been recreated as root! Sure enough!
Solution: Using a terminal as root, type:
chmod 0777 /tmp
This gives GDM permission to write to the /tmp directory when you log in as a mere user.
Note that this will only work if you have an active root account. If not, you may have to log in at a failsafe terminal (rescue mode from Grub) and do it from that command line.