dp-有限与Noetherian NIP整环

NIP整环与拓扑域之间的转换

Translating between NIP integral domains and topological fields

该研究证明NIP域上的可定义环拓扑与NIP整环紧密关联，初等等价下NIP拓扑域源自NIP整环，据此得出NIP域可定义环拓扑的局部有界性等性质，并为NIP局部环henselian猜想提供等价表述。

Henselianity in NIP $\mathbb{F}_p$-algebras

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is NIP. Suppose $R$ is additionally an $\mathbb{F}_p$-algebra. Then $R$ is a finite product of Henselian local rings. Suppose in addition that $R$ is integral. Then $R$ is a Henselian local domain, whose prime ideals are linearly ordered by inclusion. Suppose in addition that the residue field $R/\mathfrak{m}$ is infinite. Then the Artin-Schreier map $R \to R$ …

Density of algebraic points on Noetherian varieties

Let $Ω\subset{\mathbb R}^n$ be a relatively compact domain. A finite collection of real-valued functions on $Ω$ is called a \emph{Noetherian chain} if the partial derivatives of each function are expressible as polynomials in the functions. A \emph{Noetherian function} is a polynomial combination of elements of a Noetherian chain. We introduce \emph{Noetherian parameters} (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila-Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any $ε>…

Reverse mathematics of rings

Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system $\mathsf{RCA}_0 + \mathsf{I}Σ_2$. Finally, we begin a systematic study of various types of integra…