Nicolas Daans

activity

20242026

collaborators

12 papers

math.NT2026

Universally defining subrings in function fields

Nicolas Daans, Philip Dittmann

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 field…

math.NT2026

Pythagoras numbers for infinite algebraic fields

Nicolas Daans, Stevan GajoviÄ, Siu Hang Man +1

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…

cs.LG2026

Humanity's Last Exam

Long Phan, Alice Gatti, Ziwen Han +1144

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achi…

math.NT2025

Existentially defining valuations in function fields over large fields

Nicolas Daans

Let $K$ be a large field such that $K [- 1]$ is not algebraically closed and $F / K$ a function field in one variable. Extending techniques and results from earlier work with Be…

math.NT2025

Decidability of polynomial equations over function fields in positive characteristic

Nicolas Daans

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 o…

math.NT2025

Existential rank and essential dimension of diophantine sets

Nicolas Daans, Philip Dittmann, Arno Fehm

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 as…