### Expertise

In my research team we study algebraic structures such as rings and fields. These structures are used to deal with arithmetic questions. Arithmetic here means Number Theory in a broad sense. In the first place we want to decide the solvability of certain polynomial equations over fields. This involves methods from algebra, algebraic geometry, algebraic number theory, topology and modeltheory, but also analysis, combinatorics and graph theory. Furthermore we try to describe the complexity of certain structures defined over fields by obtaining bounds for the number of parameters that are needed for describing them.

## Constructive proofs for quadratic forms. 01/10/2023 - 30/09/2027

#### Abstract

We investigate the algebraic theory of quadratic forms from a constructive perspective. To this aim, we examine several known existence statements from the theory of quadratic forms over fields. For many such statements which assert the existence of a certain representation of a given object like a quadratic form or a central simple algebra, general principles from mathematical logic tell us that they should be accessible by a constructive proof, but the currently known proofs are not constructive and in particular do not give bounds on the parameters of the representation. This challenge is intriguing from the point of view of classical algebra as well as from that of constructive mathematics. The project will contribute to connect these two research areas, which promises a high potential for innovation.

#### Project type(s)

• Research Project

## First-order definitions in rings via quadratic form methods. 01/10/2020 - 30/09/2022

#### Abstract

In mathematics, a ring is a collection of objects which one can add, subtract and multiply sort of like we are used to with numbers. For example, the collection of integers (…, -2, -1, 0, 1, 2, …) forms a ring, since they can be added, subtracted or multiplied to form new integers. In our research, we intend to study techniques which might establish relationships between the computational complexity of different rings. While we find this interesting in its own right, it is further motivated by questions about which parts of mathematics can be 'automatized', in the sense that one can write a computer program to solve certain types of mathematical problems. A notorious example of this is the following, stated in some form by David Hilbert in 1900: to find an algorithm which tells you whether a given equation has a solution or not. As it turned out, what such a program might look like and even whether it exists depends heavily not only on the sorts of equations one considers, but also on what kinds of solutions one allows: only integers? Also fractions of integers? What about irrational numbers like pi? For integers, it is known that such a program cannot possibly exist; the problem cannot be automatized. On the other hand, a program is known in the case one allows all real numbers. For fractions of integers, the question remains unsolved. The techniques we will consider might bring us closer to an answer, for example by relating the complexity of the fractions and the integers.

#### Project type(s)

• Research Project

## Reduction theory of arithmetic surfaces and applications to quadratic forms over algebraic function fields 15/07/2020 - 14/07/2021

#### Abstract

The project is about the study of two groups which are naturally assosciated to a field and which contain different information on the arithmetic of the field, related to quadratic form theory. One is the quotient of the group of nonzero sums of squares modulo the subgroup of sums of two squares, the other one is the so-called Kaplansky radical. These two groups are studied in a context of arithmetic geometry, namely for algebraic function fields over a complete discretely valued field. Such a field can be viewed as the function field of an arithmetic surface over a discrete valuation ring. Classical reduction theory of algebraic curves relates our problem to the study of the special fiber of this arithmetic surface, which is an algebraic curve over the residue field. The reduction curve is connected, and the analysis of the genera of its irreducible components and of the intersections of these components relates our problems (on the two specific groups) to a combinatorial problem on the associated intersection graph. The aim of the PhD project of Gonzalo Manzano Flores in this context is in particular to obtain interesting examples of curves which illustrate how big the two groups can be under given conditions on the genus of the function field. We are searching for such examples over very specific complete discretely valued fields, namely Laurent series fields in one variable over the real and over the complex numbers, R((t)) and C((t)). The PhD candidate has obtained results on the first group (sums of squares modulo sums of two squares) in the case of a function field of a hyperelliptic curve over the field R((t)). These examples show that the upper bound for the size of this group obtained in Becher, Van Geel, Sums of squares in function fields of hyperelliptic curves. Math. Z. 261 (2009): 829–844, is best possible in the case of the function field of a hyperelliptic curve of any even degree with a nonreal function field. On the other hand, Gonzalo Manzano Flores also found evidence that this bound can be slightly improved in the case of a real function field, and that the improved bound then is optimal. The PhD candidate is currently writing up these results, which will cover a substantial part of his thesis. Complementary to this topic, we started in 2019 to work on the Kaplansky radical for function fields of arithmetic surfaces, and obtained first results and additionally evidence for some intriguing relations to the other problem. The Kaplansky radical was introduced in the 1970's by C. Cordes as a way to generalise a couple of results in quadratic form theory by substituting this group for the subgroup of squares in the field. However, it took a while until examples of fields with a nontrivial Kaplansky radical were discovered, in the 1980's by M. Kula. In the last two decades, a series of deep results on local-global principles have come up in quadratic form theory, which hold over function fields of arithmetic surfaces, and it has turned out that there is often in these situations a failure of the local-global principle in dimension 2. It was described in Becher, Leep, The Kaplansky radical of a quadratic field extension, Journal of Pure and Applied Algebra 218 (2014), 1577-1582 that this failure in dimension 2 is directly expressed by the Kaplansky radical of the corresponding function field. The aim of the collaboration with Gonzalo Manzano Flores and his supervisor at USACH, prof. David Grimm, during the period of funding will be to complement this observation by a variety of generic examples, in particular for hyperelliptic curves over C((t)). These results together with the topic on sums of squares shall lead in the academic year 2020/21 to a completion of the PhD of Gonzalo Manzano Flores, according the dubble degree convention between USACH and UA, with myself and prof. David Grimm as promotors.

#### Project type(s)

• Research Project

## First-order definitions in rings via quadratic form methods. 01/10/2018 - 30/09/2020

#### Abstract

In mathematics, a ring is a collection of objects which one can add, subtract and multiply sort of like we are used to with numbers. For example, the collection of integers (…, -2, -1, 0, 1, 2, …) forms a ring, since they can be added, subtracted or multiplied to form new integers. In our research, we intend to study techniques which might establish relationships between the computational complexity of different rings. While we find this interesting in its own right, it is further motivated by questions about which parts of mathematics can be 'automatized', in the sense that one can write a computer program to solve certain types of mathematical problems. A notorious example of this is the following, stated in some form by David Hilbert in 1900: to find an algorithm which tells you whether a given equation has a solution or not. As it turned out, what such a program might look like and even whether it exists depends heavily not only on the sorts of equations one considers, but also on what kinds of solutions one allows: only integers? Also fractions of integers? What about irrational numbers like pi? For integers, it is known that such a program cannot possibly exist; the problem cannot be automatized. On the other hand, a program is known in the case one allows all real numbers. For fractions of integers, the question remains unsolved. The techniques we will consider might bring us closer to an answer, for example by relating the complexity of the fractions and the integers.

#### Project type(s)

• Research Project

## Bounds in quadratic form theory. 01/10/2015 - 30/09/2016

#### Abstract

Quadratic forms are homogeneous polynomials of degree 2. They play an important role in modern research as special cases of various mathematical structures. Their explicit aspects make them suitable for applications. In the 1930s Ernst Witt classified quadratic forms over an arbitrary field (an algebraic structure like the rational or real numbers) by introducing the so-called Witt ring. Around the same time a similar link was drawn regarding central simple algebras, which can be classified using the Brauer group. A connection between these two algebraic structures is given by associating to a quadratic form its Clifford algebra. Merkurjev's Theorem from 1981 describes the connection. In theory, it characterizes when two forms have the same Clifford algebra and it discribes the algebras which are equivalent to Clifford algebras. The first aim of this project is to link the problem on the number of generators of the third power of the fundamental ideal in the Witt ring to Merkurjev's Theorem. Secondly, we wish to search for a more explicit proof of Merkurjev's Theorem. Thirdly, we want to make use of the fact that the central simple algebras characterized by Merkurjev's Theorem carry involutions. Further, we want to study pairs of quadratic forms in order to study the isotropy (existence of a solution) of quadratic forms over the rational function field over an arbitrary field. Finally, we wish to address the first two problems using axiomatic theory of Witt rings.

#### Project type(s)

• Research Project

## Explicit Methods in Quadratic Form Theory. 01/01/2014 - 31/12/2018

#### Abstract

The project addresses problems in Quadratic Form Theory motivated by major results obtained recently in this area. It emphasizes the use of explicit and computational approaches to obtain new results as well as refinements of results, in particular on statements about the existence of a certain representation of an object such as a quadratic form or an algebra with involution with certain properties (e.g. in terms of ramification or values for invariants). It further strives for providing criteria helping to decide for special fields whether certain forms satisfy a local-global principle or to bound certain field invariants.

#### Project type(s)

• Research Project

## The Pfister Factor Conjecture in Characteristic Two. 01/01/2014 - 31/05/2015

#### Abstract

This project represents a formal research agreement between UA and on the other hand the client. UA provides the client research results mentioned in the title of the project under the conditions as stipulated in this contract.

#### Project type(s)

• Research Project

## New methods in field arithmetic and quadratic form theory. 01/10/2013 - 30/09/2017

#### Abstract

Three different approaches lead recently to the solution of a longstanding open problem in algebra, namely to prove that the u-invariant of the function field of a curve over a p-adic number field is equal to eight. The u-invariant of a nonreal field is the smallest integer n such that every quadratic form in more than n varaibles over the field has a nontrivial zero. The different approaches are: (i) a combinatorial approach giving a far stronger result on systems of quadratic forms even over finitely generated extensions over any local number field; (ii) an approach using Galois cohomology and the construction of elements therein with given ramification; (iii) an approach called 'Field Patching' leading to new local-global principles for isotropy of quadratic forms over function fields of curves over a complete discrete valued field. Any of the three aproaches leads to more general results that so far are not attainable by the other two methods, for example (ii) and (iii) do currently not apply when p=2. The proposed doctoral research project strives for a comparative analysis of the three methods and a better understanding of their strengths and limitations. As an application it is expected that the u-invariant as well as related field invariants for other fields can be determined in this manner.

#### Project type(s)

• Research Project