某些伪代数闭赋值域上的量词消去

In the language of the classical, Weil-style algebraic geometry, a field is said to be pseudo-algebraically closed if every absolutely irreducible algebraic variety over that field has a rational point. Pseudo-finite fields, initially studied by Ax, and separably closed fields are typical examples of pseudo-algebraically closed fields. The study of pseudo-algebraically closed fields has always been a very active area of model theory since Ax started his work on pseudo-finite fields, and has various applications to other subjects like number theory and algebraic dynamics. It is often observed that valuations are useful tools in the investigation of many field-theoretic problems. Recently, algebraic geometer Kollár proved that algebraic varieties over pseudo-algebraically closed fields have some sort of denseness property with respect to nontrivial valuations over the fields. In the author's experience working with nontrivially valued separably closed fields, he thinks that this particular result of Kollár would be crucial and very helpful to the model theoretic investigation of pseudo-algebraically closed valued fields. Based on the well-known results on quantifier elimination for certain classes of pseudo-algebraically closed fields, the goal of this project is to study the quantifier elimination problems on these classes of pseudo-algebraically closed fields with valuations. This will provide a solid foundation for the model theoretic investigation of pseudo-algebraically closed fields with valuations.

使用经典的Weil的代数几何语言，如果一个域上的任何一个定义在这个域上的绝对不可约数代数簇在该域上都有有理点，那么我们就称这个域是一个伪代数闭域。经典的伪代数闭域的例子是Ax研究的伪有限域和所有的可分闭域。对于伪代数闭域的研究在Ax对伪有限域的研究以后就一直比较活跃，在数学的其他分支，例如数论和代数动力系统上都有重要应用。一个域上面的赋值结构的研究通常对研究这个域本身具有本质性的帮助。关于带有非平凡赋值的伪代数闭域的研究，近年来代数几何学家Kollár证明了一个重要的稠密性质。结合本申请者在带有非平凡赋值的可分闭域的研究的经验来看，这个稠密性质将为伪代数闭赋值域的模型论的研究起到本质性的帮助。因此本项目的任务就是从量词消去这个角度出发，建立在已知的某些伪代数闭域上的量词消去的结果之上，研究这一些特殊的伪代数闭赋值域上的量词消去问题。这将为以后的伪代数闭赋值域的模型论的研究以及应用建立基础。