Nicolas Daans
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Decidability of polynomial equations over function fields in positive characteristic Open
Let $K$ be a field of positive characteristic with no algebraically closed subfield. Let $F$ be a function field over $K$ and $t \in F$ transcendental over $K$. Refining a result of Eisentr{ä}ger and Shlapentokh, we show that there is no a…
Most totally real fields do not have universal forms or the Northcott property Open
We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a theorem on the number of…
The u-invariant of function fields in one variable Open
The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. In this article we obtain a bound on the u-invariant of function fields in one variable over a henselian valued field with arbitra…
Pythagoras numbers for infinite algebraic fields Open
We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally r…
Universal quadratic forms and Northcott property of infinite number fields Open
We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals , then the set of totally positive integers in the extension does not have the Northcott property. In particular, t…
Most totally real fields do not have universal forms or Northcott property Open
We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a new theorem on the numbe…
Universally defining subrings in function fields Open
We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of $…
Uniform existential definitions of valuations in function fields in one variable Open
We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$. We show that, for any such function field, Hilbert's 10th Problem has a…
Universal quadratic forms and Northcott property of infinite number fields Open
We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In p…
The Pythagoras number of a rational function field in two variables Open
We prove that every sum of squares in the rational function field in two variables $K(X,Y)$ over a hereditarily pythagorean field $K$ is a sum of $8$ squares. More precisely, we show that the Pythagoras number of every finite extension of …
Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers Open
We show that for a global field $K$, every ring of $S$-integers has a universal first-order definition in $K$ with $10$ quantifiers. We also give a proof that every finite intersection of valuation rings of $K$ has an existential first-ord…
Existential rank and essential dimension of diophantine sets Open
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.
Universally defining finitely generated subrings of global fields Open
It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…
Universally Defining Finitely Generated Subrings of Global Fields Open
It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…
Universally defining finitely generated subrings of global fields Open
It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…