K[[t_1，...，t_m]][x_1，.，x_n]^s
Standard Bases in K[[t_1,...,t_m]][x_1,...,x_n]^s
摘要：
本文研究了K[[t_1，…，t_m]][x_1，.，x_n]分别关于t-局部单项序的局部化。主要步骤是证明存在一个带余除法推广和结合的Grauert和莫拉的除法定理。其他的一切都自然地翻译。设m=0或n=0，作为特殊情况，分别得到了幂级数环的多项式环的标准基.然后，我们应用这种技术表明，t-初始理想的一个理想的Puietrix系列领域可以从一个标准的基础上读取其发电机。这是一个重要的步骤，在建设性的证明，每一点在热带品种这样一个理想承认一个提升。
英文摘要：
In this paper we study standard bases for submodules of K[[t_1,...,t_m]][x_1,...,x_n]^s respectively of their localisation with respect to a t-local monomial ordering. The main step is to prove the existence of a division with remainder generalising and combining the division theorems of Grauert and Mora. Everything else then translates naturally. Setting either m=0 or n=0 we get standard bases for polynomial rings respectively for power series rings as a special case. We then apply this technique to show that the t-initial ideal of an ideal over the Puiseux series field can be read of from a standard basis of its generators. This is an important step in the constructive proof that each point in the tropical variety of such an ideal admits a lifting.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
