We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Uniform existential interpretation of arithmetic in rings of functions of positive characteristic.
- Authors
Pasten, Hector; Pheidas, Thanases; Vidaux, Xavier
- Abstract
We show that first order integer arithmetic is uniformly positive-existentially interpretable in large classes of (subrings of) function fields of positive characteristic over some languages that contain the language of rings. One of the main intermediate results is a positive existential definition (in these classes), uniform among all characteristics p, of the binary relation ' $y=x^{p^{s}}$ or $x=y^{p^{s}}$ for some integer s≥0'. A natural consequence of our work is that there is no algorithm to decide whether or not a system of polynomial equations over $\mathbb {Z}[z]$ has solutions in all but finitely many polynomial rings $\mathbb {F}_{p}[z]$. Analogous consequences are deduced for the rational function fields $\mathbb {F}_{p}(z)$, over languages with a predicate for the valuation ring at zero.
- Subjects
ARITHMETIC functions; GROUP theory; MATHEMATICS theorems; DIOPHANTINE equations; POLYNOMIAL rings; FREE algebras
- Publication
Inventiones Mathematicae, 2014, Vol 196, Issue 2, p453
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-013-0472-1