Finding generic points

Date
1987
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Abstract

Generic points are classically used in algebraic geometry to, among other things, prove Hilbert's Nullstellensatz. With his development of Nonstandard Analysis, Abraham Robinson used the concept of a generic point to provide a rather simple proof of Ruckert's Nullstellensatz. His proof is recounted here in detail, with various clarifications, and then similar results about other "generic points" are explored, by applying nonstandard methods to Algebra. In the first two chapters, fundamental notions in Model Theory are presented along with basic model theoretic results about fields. These results are used in chapter three to prove the strong form of Hilbert's Nullstellensatz, and the language of Model Theory is later used to formulate the basic notions and results in Non standard Analysis. In chapter four, Ruckert's Nullstellensatz is presented without proof, and in chapter five the basic notions of Nonstandard Analysis are presented; Superstructures, the Transfer Principle, the Concurrency Principle and Enlargements. Chapter 6, entitled "Concurrency and Fields," develops the notion of a specific type of generic point, namely a hyperalgebraic primitive element for field extensions. This concept generalizes the classical notion of a primitive element in Field Theory, and here Nonstandard Algebra is first studied- The Concurrency Principle is used to find a single element ce*E for which E[less than or equal to]*F(c), whenever E is an algebraic extension over F. The seventh chapter presents A. Robinson's nonstandard proof of Ruckert's Nullstellensatz, along with certain notions of Nonstandard Analysis which are useful in this proof. The proof rests on finding a generic point for an ideal. This approach is futile in a classical setting, since the standard version of the theorem involves set and function germs, not real sets of points or functions. Chapter 8 explores more applications of the Concurrency Principle from Non standard Analysis. In particular, it sharpens the results of chapter 6 when applied to the algebraic closure of the rationals in the complex numbers- A hyperalgebraic primitive element is found in the monad of any nonstandard rational number.

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Keywords
Geometry, Algebraic, Nonstandard mathematical analysis
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