幂等半域上的多项式
Polynomials over idempotent semifields
AI总结：
研究幂等半域上单变量多项式及其因式分解，不假设幂等半域全序，确定多项式及相关函数分解为线性因子的条件，刻画代数闭幂等半域，证明完备幂等半域是代数闭的，并联系相关性质与方程解的存在性。
AI中文摘要：
我们研究系数在幂等半域中的单变量多项式及其因式分解。与该主题的大多数现有工作不同，我们不假设所考虑的幂等半域是全序的。我们特别确定多项式何时分解为线性因子，以及其相关多项式函数何时如此。这些结果使我们能够代数地刻画代数闭幂等半域——即每个多项式函数都能分解的半域。我们特别证明了每个完备幂等半域都是代数闭的。我们还将代数闭性与可预根性和可根性的性质以及多项式方程或不等式的解的存在性联系起来。
英文摘要：
We study univariate polynomials with coefficients in an idempotent semifield and their factorization. We do not assume the idempotent semifield under consideration to be totally ordered, in contrast with most of the existing work on this topic. We notably determine when a polynomial splits into linear factors, and when its associated polynomial function does so. These results lead us to characterize algebraically closed idempotent semifields -- those in which every polynomial function splits. We prove in particular that every complete idempotent semifield is algebraically closed. We also relate algebraic closedness to the properties of preradicability and radicability and to the existence of solutions to polynomial equations or inequalities.
