Model Theory and Cell Decomposition for Valued Fields with Analytic Structure

Lipshitz和Robinson建议继续合作研究具有分析结构的非阿基米德值场的模型理论。代数闭值场上的闭球解析几何，称为仿射刚性解析几何，是由Tate、Remmert等人在20世纪60年代和70年代发展起来的。Lipshitz和Robinson之前已经将仿射几何扩展到空旷球上的相对仿射几何，他们称之为准仿射。他们运用这一理论证明了刚性解析量词消去和量词简化定理，这些定理产生了关于由刚性解析函数及其图像可定义的集合结构的精确信息。他们建议继续调查。特别地，剩下的一个中心问题是在仿射刚性解析范畴中是否存在量词消去定理。他们还建议继续他们最近开始的与Raf Cluckers的合作，研究具有解析结构的（等特征零）值域的一致细胞分解定理，将pas的一致代数细胞分解扩展到具有解析结构的（等特征零）值域的情况。在离散值情况下，解析结构由仿射幂级数提供；在非离散值情况下，解析结构由拟仿射幂级数提供。关键是将经典的mittag - leffler分解推广到具有解析结构的值域中具有系数的广义环上的“解析”函数，该解析结构不一定是完全的或代数封闭的。Pas的代数细胞分解定理给出了关于p进函数的合理性在素数p上一致的结果。这项研究将把这一理论扩展到分析范畴，也将应用于动机整合。研究中使用的方法来自模型论、刚性解析函数论、交换代数和代数几何。实数的阿基米德性质是任何两个非零实数都是相称的；也就是说，有一个整数倍的第一个，其大小超过另一个的大小。数学的各个分支，例如，数论和代数几何，非阿基米德领域出现。这些域的大小概念不满足阿基米德性质：元素和的大小不大于和的最大项。这种域的一个自然例子是在域k中系数的洛朗级数域（一个变量的幂级数，最多有有限个负指数）。系数场的非零元素都有单位大小，变量的大小被认为很小（比如1/2）。幂级数满足隐函数定理，这是一个联系数量级和代数结构的中心性质。其他非阿基米德域满足亨塞尔引理，隐函数定理的推广。这些域的大量代数结构被编码在剩余域的结构中（单位大小的元素的环，是较小元素的理想模，在本例中是K）和值群（出现的大小的集合，在本例中是2的所有整数幂）。这些字段上的可定义子集通常可以分解为有限多个特别简单的部分，称为单元格。这种分解在计算数论和几何环境中出现的各种积分时非常有用。Lipshitz和Robinson建议将代数范畴内已知的非阿基米德细胞分解结果和应用扩展到解析范畴；即，对于满足Hensel引理的非阿基米德场，除多项式函数外，还定义了一类自然的解析函数。

Lipshitz and Robinson propose to continue their collaborative investigationinto the model theory of non-Archimedean valued fields with analyticstructure. The analytic geometry of closed balls over an algebraicallyclosed valued field, termed affinoid rigid analytic geometry, was developedin the 1960s and 70s by Tate, Remmert and others. Lipshitz and Robinson havepreviously extended affinoid geometry to the setting, which they termquasi-affinoid, of relative affinoid geometry over an open ball. Theyapplied that theory to prove rigid analytic quantifier elimination andquantifier simplification theorems that yield precise information about thestructure of sets definable by rigid analytic functions and their images.They propose to continue this investigation. In particular, a centralremaining question is whether there is a quantifier elimination theorem inthe affinoid rigid analytic category. They also propose to continue theirrecently-begun collaboration with Raf Cluckers into uniform celldecomposition theorems for (equicharacteristic zero) valued fields withanalytic structure, extending the uniform algebraic cell decomposition ofPas to the case of (equicharacteristic zero) valued fields with analyticstructure. In the discretely valued cases, the analytic structure isprovided by affinoid power series and in the non-discretely valued case byquasi-affinoid power series. Key is a generalization of the classicalMittag-Leffler decomposition to 'analytic' functions on a generalizedannulus with coefficients in a valued field with analytic structure that isnot necessarily complete or algebraically closed. Pas's algebraic celldecomposition theorem gives results uniform in the prime p about therationality of p-adic zeta-functions. This investigation will extend thattheory to the analytic category and will also have applications to motivicintegration. The methods to be used in the investigation come from modeltheory, rigid analytic function theory, commutative algebra and algebraicgeometry.The Archimedean property of the real numbers is the fact that any twononzero real numbers are commensurate; that is, there is an integer multipleof the first the magnitude of which exceeds the magnitude of the other. Invarious branches of mathematics, e.g., number theory and algebraic geometry,non-Archimedean fields arise. These are fields with a notion of magnitudethat does not satisfy the Archimedean property: the magnitude of a sum ofelements is no larger than the largest term of the sum. A natural example ofsuch a field is the field of Laurent series (power series in one variablewith at most finitely many negative exponents) with coefficients in a fieldK. The nonzero elements of the coefficient field all have unit magnitude,and the magnitude of the variable is considered to be small (1/2, say.)Power series satisfy the implicit function theorem, a central propertylinking the magnitude and the algebraic structure. Other non-Archimedeanfields satisfy Hensel's Lemma, a generalization of the implicit functiontheorem. A great deal of the algebraic structure of such fields is coded inthe structures of the residue field (the ring of elements of unit magnitudemodulo the ideal of smaller elements; in the example, K) and the value group(the set of magnitudes that occur; in the example, all integer powers of 2.)Definable subsets over such fields can often be decomposed into finitelymany particularly simple pieces, called cells. This decomposition is veryuseful in the evaluation of various integrals that arise in number-theoreticand geometric contexts. Lipshitz and Robinson propose to extendnon-Archimedean cell decomposition results and applications, which are knownin the algebraic category, to the analytic category; i.e. to non-Archimedeanfields satisfying Hensel's Lemma on which, in addition to the polynomialfunctions, a natural class of analytic functions is defined.