Roots of polynomials over semirings and hyperfields

Abstract

We lay the groundwork for the theory of roots of polynomials over semirings and hyperfields, employing a property of a ``surpassing relation'' $⪯$ on semirings and hyperfields that we call strong $⪯$-reversibility. There are three kinds of roots generalizing roots in classical algebraic theory --- ``null roots,'' ``$⪯$-roots,'' and ``factor-roots.'' The theory works best when all null roots are also factor-roots. The ensuing results include the fundamental theorem of algebra for pairs, asserting that tangible polynomials with enough roots ``$⪯$-split,'' sometimes uniquely, into linear factors. We also examine to what extent polynomials are determined by their null roots. Next, we obtain $⪯$-roots over extension systems, and construct integrally closed systems over various base systems, including hyperfield systems over a group. The point is that whereas hyper-polynomials behave poorly, polynomial systems remain strongly $⪯$-reversible, so one can compose chains of $⪯$-finitely spanned extensions. Next, we introduce real pairs, focusing on square roots and quadratic closure. In the appendix we recast the theory by considering equivalence classes of tangible polynomials which agree on almost all substitutions.

arXiv comment32 pages A third kind of root is studied, and a section on real roots and quadratic extensions was added. The theorems about extensions are clarified