Abstract:We continue our investigation of roots of polynomials over semirings and hyperfields, via ``pairs'' with a surpassing relation $\preceq,$ employing properties which we call $\preceq$-reversibility and strong $\preceq$-reversibility. Strong $\preceq$-reversibility yields a uniform notion of root. The ensuing results include a fundamental theorem of algebra for pairs, that tangible polynomials with enough roots ``$\preceq$-split,'' at times uniquely, into linear factors over a suitable finite extension of pairs. We also examine to what extent polynomials are determined by their null roots.
Finally, we obtain null roots of monic polynomials over extension pairs, providing a construction of integrally closed pairs over strongly $\preceq$-reversible pairs (including hyperfield pairs), and over zero sum free semirings.
From: Louis Rowen [view email]
[v1]
Thu, 11 Jun 2026 13:22:58 UTC (31 KB)
[v2]
Fri, 26 Jun 2026 11:34:33 UTC (32 KB)
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