Karim Zahidi

Elliptic divisibility sequences and undecidable problems about rational points

Abstract Julia Robinson has given a first-order definition of the rational integers ℤ in the rational numbers ℚ by a formula (∀∃∀∃) (F = 0) where the ∀-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the Σ5-theory of ℚ is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of ℤ in ℚ with quantifier complexity ∀∃, involving only one universally quantified variable. This improves the complexity of defining ℤ in ℚ in two ways, and implies that the Σ3-theory, and even the Π2-theory, of ℚ is undecidable (recall that Hilberts Tenth Problem for ℚ is the question whether the Σ1-theory of ℚ is undecidable). In short, granting the conjecture, there is a one-parameter family of hypersurfaces over ℚ for which one cannot decide whether or not they all have a rational point. The conjecture is related to properties of elliptic divisibility sequences on an elliptic curve and its image under rational 2-descent, namely existence of primitive divisors in suitable residue classes, and we discuss how to prove weaker-in-density versions of the conjecture and present some heuristics.

On Diophantine sets over polynomial rings

We prove that the recursively enumerable relations over a polynomial ring R[t], where R is the ring of integers in a totally real number field, are exactly the Diophantine relations over R[t].

Model Theory with Applications to Algebra and Analysis: Decision problems in Algebra and analogues of Hilbert's tenth problem

Introduction One of the first tasks undertaken by Model Theory was to produce elimination results, for example methods of eliminating quantifiers in formulas of certain structures. In almost all cases those methods have been effective and thus provide algorithms for examining the truth of possible theorems. On the other hand, Godels Incompleteness Theorem and many subsequent results show that in certain structures, constructive elimination is impossible. The current article is a (very incomplete) effort to survey some results of each kind, with a focus on the decidability of existential theories, and ask some questions at the intersection of Logic and Number Theory. It has been written having in mind a mathematician without prior exposition to Model Theory. Our presentation will consist of four parts. Part A deals with positive (decidability) results for analogues of Hilberts tenth problem for substructures of the integers and for certain local rings. Part B focuses on the ‘parametric problem’ and the relevance of Hilberts tenth problem to conjectures of Lang. Part C deals with the analogue of Hilberts tenth problem for rings of Analytic and Meromorphic functions. Part D is an informal discussion on the chances of proving a negative (or could it be positive?) answer to the analogue of Hilberts tenth problem for the field of rational numbers.

On the u-invariant of p-adic function fields

ABSTRACT We show that a quadratic form defined over the rational function field ℚ(x 1 , …, x n ) of dimension at least 4.2 n + 1 is isotropic over all fields ℚ p (x 1 , …, x n ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown.