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。代数闭值域上闭球的解析几何，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以前已将仿射几何推广到开球上相对仿射几何的设置，他们称之为准仿射几何。他们应用这一理论证明了刚性解析量词消去法和量词简化定理，这些定理给出了关于可由刚性解析函数及其像定义的集合的结构的精确信息。他们建议继续这项研究。特别是，一个中心的剩余问题是在仿射刚性分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，得到具有解析结构的(等特征零)值场的一致胞胞分解定理，将PA的一致代数胞分解推广到具有解析结构的(等特征零)值场的情况。在离散值情形下，解析结构由仿射幂级数给出，在非离散值情形下，解析结构由拟仿射幂级数给出。关键是将经典的Mittag-Leffler分解推广到广义环上的“解析”函数，其系数在具有解析结构的值域中，不必是完全的或代数闭合的。PAS的代数胞格分解定理给出了p-进Zeta-函数的相关性在素数p内一致的结果。这项研究将把这一理论扩展到分析范畴，并将应用于动机整合。实数的阿基米德性质是指任意二个零实数是公度的，即存在一个第一个数的大小超过另一个数大小的整数倍。数学的不变分支，例如数论和代数几何，非阿基米德领域应运而生。这些域的量值概念不满足阿基米德性质：元素和的量值不大于该和的最大项。这样的场的一个自然例子是具有域K中的系数的洛朗级数(一元幂级数，至多有有限多个负指数)的场。系数域的非零元素都有单位量级，变量的量级被认为是小的(比如1/2)。幂函数级数满足隐函数定理，这是连接量级和代数结构的一个中心性质。其他非阿基米德域满足Hensel引理，这是隐函数定理的推广。这样的域的大量代数结构被编码在剩余域(单位幅值的元素环模小元素的理想；在例子中是K)和值组(出现的幅值的集合；在例子中是所有2的整数次幂)的结构中。这样的域上的可定义子集通常可以分解成有限多个特别简单的片段，称为单元。这种分解在计算数论和几何背景下出现的各种积分时非常有用。Lipshitz和Robinson建议将代数范畴中已知的非阿基米德胞元分解结果及其应用推广到解析范畴，即满足Hensel引理的非阿基米德域，在该引理上，除了多项式函数外，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.