Model Theory and Cell Decomposition for Valued Fields with Analytic Structure

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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.