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Nicolas Daans

Publications and source records attributed to Nicolas Daans.

14 recordsLinked to original sources
arXiv2025

Existentially defining valuations in function fields over large fields

Let $K$ be a large field such that $K[\sqrt{-1}]$ is not algebraically closed and $F/K$ a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on $F$ containing $K$ is existentially definable in the language of rings with parameters from $F$. As a consequence, using a known reduction technique, we obtain the undecidability of the existential theory of $F$ in the language of rings with appropriately chosen parameters.

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arXiv2025

Decidability of polynomial equations over function fields in positive characteristic

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{\"a}ger and Shlapentokh, we show that there is no algorithm which, on input a polynomial $f \in \mathbb{Z}[t][X_1, \ldots, X_n]$, determines whether $f$ has a zero in $F^n$. To this end, we revisit and partially extend several recent results from the literature on existential definability in function fields.

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arXiv2025

The u-invariant of function fields in one variable

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 arbitrary value group and with residue field of characteristic different from 2. This generalises a theorem due to Harbater, Hartmann and Krashen and its extension due to Scheiderer. Their result covers the special case where the valuation is discrete. We further give a new proof of a theorem due to Parimala and Suresh bounding by 8 the u-invariant of a function field in one variable over any henselian discretely valued field of characteristic 0 with perfect residue field of characteristic 2.

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arXiv2025

Pythagoras numbers for infinite algebraic fields

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 real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.

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arXiv2024

Most totally real fields do not have universal forms or Northcott property

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 number of square classes of totally positive units represented by a quadratic lattice of a given rank.

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arXiv2024

Failures of integral Springer's Theorem

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove that this phenomenon happens infinitely often, and, conversely, establish finiteness results about the situation when the quadratic form is fixed.

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arXiv2024

Universally defining subrings in function fields

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 $S$-integers is always a diophantine set. As a technical tool, we use a reciprocity exact sequence for quadratic Witt groups in function fields over almost arbitrary base fields (of any characteristic), which is new and of potentially independent interest.

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arXiv2024

Linkage of Pfister forms over semi-global fields

We study linkage of $(d+1)$-fold quadratic Pfister forms over function fields in one variable over a henselian valued field of 2-cohomological dimension $d$. Specifically, we characterise this property in terms of linkage of quadratic Pfister forms over function fields over the residue field of the henselian valued field; in full generality in characteristic different from 2, and for most complete discretely valued fields in characteristic 2. As an application, we obtain a proof that $(d+2)$-fold quadratic Pfister forms over function fields in one variable over a $d$-dimensional higher local field are linked.

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arXiv2023

Uniform existential definitions of valuations in function fields in one variable

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 negative answer, the valuation rings containing $K$ are uniformly existentially definable, and finitely generated integrally closed $K$-subalgebras are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic.

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arXiv2023

Universal quadratic forms and Northcott property of infinite number fields

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 particular, this implies that no universal form exists over the compositum of all totally real Galois fields of a fixed prime degree over $\mathbb{Q}$. Further, by considering the existence of infinitely many square classes of totally positive units, we show that no classical universal form exists over the compositum of all such fields of degree $3d$ (for each fixed odd integer $d$).

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arXiv2023

The Pythagoras number of a rational function field in two variables

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 $K(X)$ is at most $5$. The main ingredients of the proof are a local-global principle for quadratic forms over function fields in one variable over a complete rank-$1$ valued field due to V. Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to L. Br\"ocker.

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arXiv2023

Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers

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-order definition in $K$ with $3$ quantifiers.

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arXiv2018

Universally defining finitely generated subrings of global fields

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 number fields and rings of $S$-integers in global function fields of odd characteristic. In this article a proof is presented which is uniform in all global fields, including the characteristic two case, where the result is entirely novel. Furthermore, the proposed method results in universal formulae requiring significantly fewer quantifiers than the formulae that can be derived through the previous approaches.

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