2026/06/30 by Louis Halle Rowen
#math.RA
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.