This article is a survey about analogues of Hilbert's Tenth Problem over various rings, especially rings of interest to number theorists and algebraic geometers. For more details about most of the topics considered here, the conference proceedings [DLPVG00] is recommended. 2. The original problem Hilbert's Tenth Problem (from his list of 23 problems published in 1900) asked for an algorithm to decide whether a diophantine equation has a solution. More precisely, the input and output of such an algorithm were to be as follows: input: a polynomial f(x 1 , . . . , x n ) having coe#cients in Z Date: February 28, 2003
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2877
|
Introduction to Automata Theory, Languages, and Computation
– Hopcroft, Ullman
- 1979
|
|
428
|
Algebraic geometry
– Hartshorne
- 1977
|
|
364
|
A Decision Method for Elementary Algebra and Geometry
– Tarski
- 1951
|
|
360
|
The arithmetic of elliptic curves
– Silverman
- 1992
|
|
232
|
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik
– Gödel
- 1931
|
|
76
|
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent
– Faltings
- 1983
|
|
29
|
Diophantine problems over local fields
– Ax, Kochen
- 1965
|
|
28
|
The decision problem for exponential Diophantine equations
– Davis, Putnam, et al.
- 1961
|
|
26
|
On definable subsets of p-adic fields
– Macintyre
- 1976
|
|
26
|
Definability and Decision Problems in Arithmetic
– Robinson
- 1949
|
|
14
|
The topology of rational points
– Mazur
- 1992
|
|
13
|
Proof of recursive unsolvability of Hilbert’s tenth problem
– Matiyasevich, Jones
- 1991
|
|
11
|
Diophantine sets over some rings of algebraic integers
– Denef, Lipshitz
- 1978
|
|
10
|
Double fibres and double covers: paucity of rational points
– Colliot-Thélène, Skorobogatov, et al.
- 1997
|
|
10
|
Hilbert’s tenth problem for rational function fields in characteristic 2
– Videla
- 1994
|
|
9
|
Topology of Diophantine sets: remarks on Mazur’s conjectures, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent
– Cornelissen, Zahidi
- 1999
|
|
9
|
Speculations about the topology of rational points: an update, Astérisque
– Mazur
- 1995
|
|
9
|
Hilbert’s tenth problem for a class of rings of algebraic integers
– Pheidas
- 1988
|
|
9
|
Hilbert’s tenth problem for fields of rational functions over finite fields
– Pheidas
- 1991
|
|
9
|
Extension of Hilbert’s tenth problem to some algebraic number fields
– Shlapentokh
- 1989
|
|
7
|
The Diophantine problem for polynomial rings and fields of rational functions
– Denef
- 1978
|
|
6
|
Diophantine undecidability of C(t1, t2
– Kim, Roush
- 1992
|
|
6
|
The Diophantineness of enumerable sets
– Matijasevič
- 1970
|
|
6
|
Diophantine classes of holomorphy rings of global fields
– Shlapentokh
- 1994
|
|
6
|
Hilbert’s tenth problem for algebraic function fields over infinite fields of constants of positive characteristic
– Shlapentokh
- 2000
|
|
5
|
A decision method for p-adic integral zeros of Diophantine equations
– Nerode
- 1963
|
|
5
|
A ring version of Mazur’s conjecture on topology of rational points
– Shlapentokh
|
|
5
|
The method of trigonometrical sums in the theory of numbers, Interscience
– Vinogradov
- 1954
|
|
4
|
Hilbert’s Tenth Problem and Mazur’s conjecture for large subrings of Q
– Poonen
|
|
4
|
The undecidability of algebraic rings and
– Robinson
- 1959
|
|
3
|
Hilbert’s Tenth Problem for algebraic function fields of characteristic 2
– Eisenträger
- 2003
|
|
3
|
An approach to rational Diophantine undecidability
– Kim, Roush
- 1992
|
|
3
|
Using elliptic curves of rank one towards the undecidability of Hilbert’s Tenth Problem over rings of algebraic integers
– Poonen
- 2002
|
|
2
|
Diophantine sets over algebraic integer rings
– Denef
- 1980
|
|
2
|
On the elementary theory of maximal normed fields, Dokl
– Erˇsov
- 1965
|
|
2
|
Undecidability of fields of rational functions over fields of characteristic 2. (Russian) Algebra i Logika 12
– Penzin
- 1973
|
|
2
|
Hilbert’s tenth problem for rings of algebraic functions in one variable over fields of constants of positive characteristic
– Shlapentokh
- 1992
|
|
1
|
The undecidability of certain fields
– Erˇsov
- 1965
|
