# Existential rank and essential dimension of diophantine sets

@inproceedings{Daans2021ExistentialRA, title={Existential rank and essential dimension of diophantine sets}, author={Nicolas Daans and Philip Dittmann and Arno Fehm}, year={2021} }

We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points associated to such a definition.

## 2 Citations

NOTES ON THE DPRM PROPERTY FOR LISTABLE STRUCTURES

- MathematicsThe Journal of Symbolic Logic
- 2021

A celebrated result by M. Davis, H. Putnam, J. Robinson, and Y. Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of…

$\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$ with 32 unknowns

- Mathematics
- 2021

Let Z be the ring of integers. Hilbert’s Tenth Problem (HTP), the tenth one of his 23 famous mathematical problems presented in the 1900 ICM, asks for an algorithm to determine for any given…

## References

SHOWING 1-10 OF 66 REFERENCES

Defining Z in Q

- Physics, Mathematics
- 2010

We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$…

Algebraic Geometry and Arithmetic Curves

- Mathematics
- 2002

Introduction 1. Some topics in commutative algebra 2. General Properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohmology 6. Sheaves of…

Further results on Hilbert’s Tenth Problem

- MathematicsScience China Mathematics
- 2020

Hilbert’s Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring ℤ of integers. This was finally…

Embedding problems over large fields

- Mathematics
- 1996

In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on…

Universal Diophantine Equation

- Mathematics, Computer ScienceJ. Symb. Log.
- 1982

Matijasevic's theorem implies the existence of a diophantine equation U such that for all x and v, x ∈ W v is also recursively enumerable, and the nonexistence of such an algorithm follows immediately from theexistence of r.e. nonrecursive sets.

NOTES ON THE DPRM PROPERTY FOR LISTABLE STRUCTURES

- MathematicsThe Journal of Symbolic Logic
- 2021

A celebrated result by M. Davis, H. Putnam, J. Robinson, and Y. Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of…

A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

- 2020

In this paper we investigate the algebraic extensions K of Q in which we cannot existentially or universally define the ring of integersOK . A complete answer to this question would have important…

Points on Curves

- PhD thesis, Princeton
- 2020

Existentially generated subfields of large fields

- MathematicsJournal of Algebra
- 2019

We study subfields of large fields which are generated by infinite existentially definable subsets. We say that such subfields are existentially generated.
Let $L$ be a large field of characteristic…