A continuous, constructive solution to Hilbert's 17th problem.
A failure of quantifier elimination.
We show that log is needed to eliminate quantifiers in the theory of the real numbers with restricted analytic functions and exponentiation.
A remark on Ax's theorem on solvability modulo primes.
Application de la logique à un problème de dimension diophantienne sur les corps
Approximation diophantienne et distances ultramétriques non standard
Büchi's problem in any power for finite fields
Definability in the extended arithmetic of ordinal numbers [Book]
Definability within structures related to Pascal’s triangle modulo an integer
Let Sq denote the set of squares, and let be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let . For every integer n ≥ 2 addition and multiplication are definable in the structures ⟨ℕ; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.
Definable sets over finite fields.
Deux propriétés décidables des suites récurrentes linéaires
Diophantine representation of Mersenne and Fermat primes
Diophantine representation of the decimal expansions of and
Diophantine undecidability for addition and divisibility in polynomial rings
We prove that the positive-existential theory of addition and divisibility in a ring of polynomials in two variables A[t₁,t₂] over an integral domain A is undecidable and that the universal-existential theory of A[t₁] is undecidable.
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Let be a one-variable function field over a field of constants of characteristic 0. Let be a holomorphy subring of , not equal to . We prove the following undecidability results for : if is recursive, then Hilbert’s Tenth Problem is undecidable in . In general, there exist such that there is no algorithm to tell whether a polynomial equation with coefficients in has solutions in .
Diophantische Gleichungen und die universelle Eigenschaft Finslerscher Zahlen.
Dirichlet theorems and prime number hypotheses of a conditional Goldbach theorem.
Disjoint covering systems and product-invariant relations
Division-ample sets and the Diophantine problem for rings of integers
We prove that Hilbert’s Tenth Problem for a ring of integers in a number field has a negative answer if satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over ). We relate division-ample sets to arithmetic of abelian varieties.
Ein System algebraisch unabhängiger Zahlen
Équations diophantiennes modulo