ON SUPERRINGS OF POLYNOMIALS AND ALGEBRAICALLY CLOSED MULTIFIELDS

The concept of multialgebraic structure - an "algebraic like" structure but endowed with multiple valued operations - has been studied since the 1930's; in particular, the concept of hyperrings was introduced by Krasner in the 1950's. Some general algebraic study has been made on multialgebras: see for instance [9] and [17]. More recently the notion of multiring have obtained more attention: a multiring is a lax hyperring, satisfying an weak distributive law, but hyperfields and multifields coincide. Multirings has been studied for applications in abstract quadratic forms theory ([12], [8]) and tropical geometry ([10]); a more detailed account of variants of concept of polynomials over hyperrings is even more recent ([10], [4]). In the present work we start a model-theoretic oriented analysis of multialgebras introducing the class of algebraically closed and providing variant proof of quantifier elimination flavor, based on new results on superring of polynomials ([4]).