Roots of polynomials over semirings and hyperfields

Abstract

We continue our investigation of roots of polynomials over semirings and hyperfields, employing a property on semiring and hyperfield ``pairs'' with a surpassing relation $\preceq,$ which we call $\preceq$-reversibility. There are two kinds of roots generalizing the classical algebraic theory, ``null roots,'' and $\preceq$-roots. The theory works best when all null roots are also $\preceq$-roots. Ensuing results include the fundamental theorem of algebra for pairs, that tangible polynomials with enough roots ``$\preceq$-split,'' at times uniquely, into linear factors. We also see that polynomials that agree on ``almost'' all null roots are ``almost'' equal. Finally, we obtain roots of integral polynomials over extension pairs, providing a construction of integrally closed pairs over hyperfields and over zero sum free semirings.