you are replacing the complete part after the first dot in the script path and replace it by .log
Good point. And here is one way of calling the script to reproduce the OP's result: 
$ ./my_script_name.pl
.log

[download]
Perhaps the OP has invoked the script differently on the 2 OS's.

In reply to Re^2: Different behavior when parsing $0 on Solaris and Win32 by toolic
in thread Different behavior when parsing $0 on Solaris and Win32 by dirac

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