论文标题
田园诗的热带扩展和贝克 - 洛斯切德多重性
Tropical Extensions and Baker-Lorscheid Multiplicities for Idylls
论文作者
论文摘要
Matthew Baker和Oliver Lorscheid在最近的一篇论文中表明，笛卡尔的迹象规则，牛顿的多边形规则都可以解释为多项式在Hyperfields上的多样性。 Hyperfields是字段的概括，它们编码符号或绝对值的算术之类的内容。通过查看此类代数上的多项式的多重性，Baker和Lorscheid表明您可以恢复笛卡尔和牛顿的规则。 在本文中，我们定义了田园诗的热带扩展。在Akian-Gaubert-Guterman的工作中，出现了与否定对称性的半段，以及Bowler-SU的工作中的超组和高场。热带扩展的示例将热带超场扩展到更高的等级，或者通过包括估值将标志的符号扩展到热带真实的超场。 本文的结果涉及多重性和热带扩展的相互作用。首先，有一个从初始形式到整个多项式的提升定理，我们将从中证明多项式的多重性等于某些初始形式的相应多重性。其次，我们表明，热带扩展可以保留所有多重性之和受该程度界定的特性。因此，我们为每个严格的高场都有此程度的界限。这给了贝克和洛尔切德提出的关于哪些Hyperfield拥有此属性的问题的部分答案。
In a recent paper, Matthew Baker and Oliver Lorscheid showed that Descartes's Rule of Signs and Newton's Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebras, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property.
