Model Theory of Fields: Decidability and Bounds for Polynomial Ideals
Lou van den Dries
Abstract:
A favorite activity of mathematicians has always been solving equations, this 'solving' in a broad sense.
Until the 20th century, the emphasis was on finding direct, algorithmic methods, which will always be of the utmost importance.
Now consider an equation for example
(*) f(x1, .., xn) = 0 (f a polynomial with rational coefficients), where solutions are required in rational numbers, a so-called Diophantine equation.
For even fairly simple Diophantine equations, algorithmic solution methods appeared not to be available or to provide little insight. In order to obtain the desired information about the solutions, for example, the solutions of (*) were studied in the p-adic fields Qp and in the field R of the real numbers. This process, called localization and completion, proves to be very useful, especially in the related field of algebraic geometry (see, for example, the 'Introduction' of [Bo2]).
One can even advantageously replace the solution in all Qp and in R by the solution in one ring, the ring A of the adÃ©les, which has Q as a subring. Now A has proven to be particularly suitable for arithmetic purposes because of its topological properties. This suitability has recently been confirmed by its model-theoretic properties: there is an effective method to check whether a given 'elementary' statement about rings is true for A, in particular. one can determine from an equation (*) whether there are solutions in A, whether there are infinitely many, etc. This result (Weispfenning, as yet unpublished) can be regarded as a summary of earlier work by A. Tarski, A. Robinson, J. Ax, S. Kochen, Ju. Erdov and P.J. Cohen.
Now the importance of A for Diophantine equations is highly dependent on: which properties of Q are reflected in A? One can e.g. say that some 'quadratic' properties of Q can be easily found in A (Hasse-Minkowski). But Q has no zero divisors and A does. O.a. these considerations have led me to study the model-theoretic aspects of the bodies discussed in chapters II and III. Typical example: consider the objects (K,<,v1, v2) with K a field, < a linear order on K, v1: K* -> Z a p-adic valuation, i.e. v1(p) = 1 and K_v1 = F_p, and v2: K* -> Z a q-adic valuation (p and q given prime numbers).
For the 'existentially closed' objects in this category, a result as described above for A appears to indeed apply (see Ch. III, (3.1)). My hope is that these existentially closed objects preserve the structure of Q better than the ring RxQpxQq.
Chapter IV is of a different nature: it solves some problems suggested by A. Robinson, see [Rob4, problem 3].