"Under signature" is a phrase commonly used to refer to a document or message that has been signed by the author or sender. Synonyms for this phrase include "signed by", "endorsed by", "authored by", and "approved by". These synonyms can be used interchangeably in legal, business, and personal contexts when referring to a document or message that has been signed. In some cases, "under signature" may also be replaced with "signed off on" or "executed by". Regardless of the synonym used, the meaning remains the same - that the document or message has been verified and approved by the appropriate party.