发表机构
Korea Institute for Advanced Study（韩国高等科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出$n$-变量定理，统一处理多种分裂线，并证明在泛平凡化、可爱对、$H$-结构、向量空间、代数闭域及泛导数等扩张下保持性，同时推广到更高元数分裂线。
AI 中文摘要
我们利用正一致性-不一致性配置和广义不可区分元，定义了一类完整一阶理论类，记为$\mathfrak{D}_{n\text{-var}}$。该类包含稳定、简单、NIP、NTP$_1$、NTP$_2$、NATP、NCTP和NBTP理论类，以及对于任意$k<\omega$不具有深度$\omega$的$(k,1,1)$-编织的理论类、对于任意$k<\omega$不具有无限$k$-网格的理论类，以及对于每个$1<k<\omega$的NPM$^{(k)}$理论类。对于任意分裂线$D\in\mathfrak{D}_{n\text{-var}}$和任意完整一阶理论$T\notin D$，我们总能找到一个公式$\varphi(x,y)$，它通过一个带索引的参数集见证$T\notin D$，使得其所需一致的实例集有一个实现，该实现在整个参数集上的代数维数为$|x|$。我们将这种形式的陈述称为$n$-变量定理，因为它们可被视为单变量定理的弱化版本。在$T$的适当假设下，我们证明$T\in D$当且仅当相应的理论属于$D$，对于(i)$T^{gt}$，(ii)$T_P$，(iii)$T^{ind}$，(iv)$T^G_K$，(v)ACF$_T$，以及(vi)$T^\delta_g$的任意完备化。这些分别是泛平凡化理论、可爱对扩张理论、$H$-结构扩张理论、带有稠密-余稠密泛$K$-子空间的向量空间理论、带有指定子域的代数闭域理论，以及代数有界域的泛导数理论。$n$-变量定理在证明(i)--(v)中至关重要。我们还引入了一个更大的类$\mathfrak{D}^h_{n\text{-var}}\supseteq\mathfrak{D}_{n\text{-var}}$，用于捕获更高元数的分裂线，如对于$1<k<\omega$的NOP$_k$、NFOP$_k$和NIP$_k$。$n$-变量定理和保持结果(ii)、(iii)、(iv)和(vi)也适用于这个更大的类，而(i)在假设${\rm acl}={\rm dcl}$下成立。
英文摘要
We define a class of classes of complete first-order theories, denoted by $\mathfrak{D}_{n\text{-var}}$, using positive consistency-inconsistency configurations and generalized indiscernibles. It contains the classes of stable, simple, NIP, NTP$_1$, NTP$_2$, NATP, NCTP, and NBTP theories, as well as the classes of theories not having a $(k,1,1)$-weave of depth $ω$ for any $k<ω$, theories not having an infinite $k$-grid for any $k<ω$, and NPM$^{(k)}$ theories for each $1<k<ω$. For any dividing line $D\in\mathfrak{D}_{n\text{-var}}$ and any complete first-order theory $T\notin D$, we can always find a formula $φ(x,y)$ witnessing $T\notin D$ with an indexed set of parameters such that the set of its instances required to be consistent has a realization whose algebraic dimension over the whole set of parameters is $|x|$. We call statements of this form $n$-variable theorems, as they may be regarded as weak versions of one-variable theorems. Under the appropriate hypotheses on $T$, we prove that $T\in D$ if and only if the corresponding theory belongs to $D$ for (i) $T^{gt}$, (ii) $T_P$, (iii) $T^{ind}$, (iv) $T^G_K$, (v) ACF$_T$, and (vi) any completion of $T^δ_g$. These are, respectively, the theories of generic trivializations, lovely pair expansions, $H$-structure expansions, vector spaces with a dense-codense generic $K$-subspace, algebraically closed fields with a distinguished subfield, and generic derivations of algebraically bounded fields. The $n$-variable theorem is essential in proving (i)--(v). We also introduce a larger class $\mathfrak{D}^h_{n\text{-var}}\supseteq\mathfrak{D}_{n\text{-var}}$, capturing higher-arity dividing lines such as NOP$_k$, NFOP$_k$, and NIP$_k$ for $1<k<ω$. The $n$-variable theorem and preservation results (ii), (iii), (iv), and (vi) also hold for this larger class, while (i) holds assuming ${\rm acl}={\rm dcl}$.
