23 citations · 34 across the 15 of their papers we have counts for
15 papers
Existentially defining valuations in function fields over large fields
Nicolas Daans
Let be a large field such that is not algebraically closed and a function field in one variable. Extending techniques and results from earlier work with Be…
Decidability of polynomial equations over function fields in positive characteristic
Nicolas Daans
Let be a field of positive characteristic with no algebraically closed subfield. Let be a function field over and transcendental over . Refining a result o…
The u-invariant of function fields in one variable
Karim Johannes Becher, Nicolas Daans, Vlerë Mehmeti
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…
Pythagoras numbers for infinite algebraic fields
Nicolas Daans, Stevan Gajović, 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…
Humanity's Last Exam
Long Phan, Alice Gatti, Ziwen Han +1168
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…
Most totally real fields do not have universal forms or Northcott property
Nicolas Daans, Vitezslav Kala, Siu Hang Man +2
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 prope…
Failures of integral Springer's Theorem
Nicolas Daans, Vítězslav Kala, Jakub Krásenský +1
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 f…
Universally defining subrings in function fields
Nicolas Daans, Philip Dittmann
We establish that all rings of -integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local field…
Linkage of Pfister forms over semi-global fields
Nicolas Daans
We study linkage of -fold quadratic Pfister forms over function fields in one variable over a henselian valued field of 2-cohomological dimension . Specifically, we chara…
Uniform existential definitions of valuations in function fields in one variable
Karim Johannes Becher, Nicolas Daans, Philip Dittmann
We study function fields of curves over a base field which is either a global field or a large field having a separable field extension of degree divisible by . We show that…
Universal quadratic forms and Northcott property of infinite number fields
Nicolas Daans, Vítězslav Kala, Siu Hang Man
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…
Sums of squares in function fields over a henselian valued field
Karim Johannes Becher, Nicolas Daans, David Grimm +2
The Pythagoras number of a field is the smallest natural number such that every sum of squares in the field is a sum of squares. Given a nondyadic henselian valued field $(…
Universally defining in with quantifiers
Nicolas Daans
We show that for a global field , every ring of -integers has a universal first-order definition in with quantifiers. We also give a proof that every finite intersec…
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…
Universally defining finitely generated subrings of global fields
Nicolas Daans
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 int…