Can you include a copy of your php.ini file?

I think this is the relevant section:
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Error handling and logging ;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

; error_reporting is a bit-field.  Or each number up to get desired error
; reporting level
; E_ALL             - All errors and warnings
; E_ERROR           - fatal run-time errors
; E_WARNING         - run-time warnings (non-fatal errors)
; E_PARSE           - compile-time parse errors
; E_NOTICE          - run-time notices (these are warnings which often result
;                     from a bug in your code, but it's possible that it was
;                     intentional (e.g., using an uninitialized variable and
;                     relying on the fact it's automatically initialized to an
;                     empty string)
; E_CORE_ERROR      - fatal errors that occur during PHP's initial startup
; E_CORE_WARNING    - warnings (non-fatal errors) that occur during PHP's
;                     initial startup
; E_COMPILE_ERROR   - fatal compile-time errors
; E_COMPILE_WARNING - compile-time warnings (non-fatal errors)
; E_USER_ERROR      - user-generated error message
; E_USER_WARNING    - user-generated warning message
; E_USER_NOTICE     - user-generated notice message
;
; Examples:
;
;   - Show all errors, except for notices
;
;error_reporting = E_ALL & ~E_NOTICE
;
;   - Show only errors
;
;error_reporting = E_COMPILE_ERROR|E_ERROR|E_CORE_ERROR
;
;   - Show all errors except for notices
;
error_reporting  = E_ALL; display all errors, warnings and notices

; Print out errors (as a part of the output).  For production web sites,
; you're strongly encouraged to turn this feature off, and use error logging
; instead (see below).  Keeping display_errors enabled on a production web site
; may reveal security information to end users, such as file paths on your Web
; server, your database schema or other information.
display_errors = On

; Even when display_errors is on, errors that occur during PHP's startup
; sequence are not displayed.  It's strongly recommended to keep
; display_startup_errors off, except for when debugging.
display_startup_errors = Off

; Log errors into a log file (server-specific log, stderr, or error_log (below))
; As stated above, you're strongly advised to use error logging in place of
; error displaying on production web sites.
log_errors = Off

; Store the last error/warning message in $php_errormsg (boolean).
track_errors = Off

; String to output before an error message.
;error_prepend_string = "<font color=ff0000>"

; String to output after an error message.
;error_append_string = "</font>"

; Log errors to specified file.
;error_log = filename

; Log errors to syslog (Event Log on NT, not valid in Windows 95).
;error_log = syslog

; Warn if the + operator is used with strings.
warn_plus_overloading = Off

However, since QUERY_STRING is a PHP variable, I think it should be defined as "" if there is no querystring specified.

in your php.ini file make sure this line is included...

register_globals = on

register globals is on.  The php.ini is the same one that came with the installer available at php.net.  We made no changes to it.

No bug here.  The default php.ini file has error_reporting set to E_ALL & ~E_NOTICE.  You have it set to E_ALL which is why you get the warning.
The proper way to code this is to use:

if(isset($QUERY_STRING)) ...

And no, $QUERY_STRING should not be set when there is no query string arguments in the URL.