I would suggest that in public you be as professional and polite to him as you can and if there is a reason to compliment him, make sure you do that. If things come down to there being such a conflict that one of you has to go, who is going to look better, the one behaving professionally or the one behaving like a five-year-old?
However, if he starts publicly presenting an untrue story about you, correct the misinformation in public. Don't let his version of events be the only one. I am always polite, but I never let anyone spread lies about me. One guy tried this with me all the way up to the CEO of the company (I had never even met him but my name came up for some assignment that he didn't want to leave his department) and I had to to take his lies apart just as publicly and in such a way that he never bothered me again. I wasn't nasty, I didn't say any untruths back and I didn't call him names (althoughI was furious and wanted to), but after he had lied about me in writing questioning my professional competence, I sent back to the CEO (and all the Senior management staff he had orginally sent the document to) a multi-page document detailing all the lies and providing proof that they were lies. He never tried that with me again. Sometimes you have to let a bully know that you are a dangerous person to try to bully. In this case I did this with the full support of my boss (who was the person who showed me the orginal document so that I could respond, he was pretty mad too.). If you have to take the guy down publicly, it is good to have your own boss's support before you do so.
Make sure to cover yourself with your boss by making sure he is aware of your contributions. Hopefully this won;t go so far you need a smack down to avoid a destroyed reputation, but in any event, making sure your boss is aware of all the good things you do helps mitigate anything he says when you are not there to hear it.