"Hinge upon" is a phrase that suggests an event or outcome depends heavily on a particular factor or decision. There are several synonyms that can be used instead of "hinge upon" such as "depend on", "rest on", "turn on", "be contingent on", "be determined by", "be influenced by", "be swayed by", "be decided by", "be governed by" and "be controlled by". These synonyms convey the same idea that a particular action or decision will have a significant impact on the result of a situation. Choosing the right synonym will depend on the context and tone of the sentence.