Probing the boundaries of the rationality problem
The solutions to polynomial equations in degree 2 can generally be parameterized, yet it is largely unknown whether this is possible for equations of degree as low as 3. The team behind the ERC-backed RationAlgic project aim to show that the solutions to polynomial equations in a certain degree and dimension range cannot be parameterized one to one, as Professor Stefan Schreieder explains.
The rationality problem in the field of algebraic geometry can be addressed by parameterizing solutions to polynomial equations, an approach which is used to understand the zero sets of these equations. Among the simplest examples is the pythagorean equation, where x 2+y2=z 2 . “Such equations have certain invariants. There is the number of variables, and then there is the degree,” explains Stefan Schreieder, Head of the Institute of Algebraic Geometry at the Leibniz University Hannover. The solutions to this equation can be parameterized, which can be thought of as essentially a means of describing the solutions. “It is possible to parameterize the pythagorean equation, and in general it is always possible to parameterize the solutions to equations in degree 2,” says Professor Schreieder. “However, it gets much more complicated to parameterize the solutions when you start looking at more complex equations with a larger degree, or with more variables.”
RationAlgic project
This topic lies at the core of Professor Schreieder’s work as Principal Investigator
of the ERC-backed RationAlgic project, in which he and his team aim to prove that the solutions to equations in a certain degree and dimension range cannot be parameterized. The degree of an equation is, broadly speaking, the highest power term within it – for example, the degree of x3+xy2 would be 3 – while the dimension is the number of variables within the
are not rational, in the sense that their solutions also cannot be parameterized.”
The aim now is to extend this work and show that equations in a certain degree and dimension range cannot be parameterized, focusing primarily on homogenous equations, where all terms have the same degree. A variety of methods are being applied in this work,
about the irrationality of varieties,” he explains. Professor Schreieder’s work in this area pre-dates the project; in 2019 he published a paper which he says represented a significant breakthrough.
“Previously people had understood the rationality problem for general equations whose degree is bounded from below by a linear bound in terms of the number of variables, i.e. on the dimension. In this degree and dimension range it was known that the solutions to general equations cannot be parameterized,” he outlines.
In his 2019 paper Professor Schreieder found a logarithmic bound, which grows substantially slower than any linear or polynomial bound. This work has opened up new insights into which types of equations can be parameterized.
“Previously, in dimension 1 million, the bound on the degree was in the region of 300,000, but this has been reduced to just 22,” says Professor Schreieder. The 2019
paper has since been refined further, and Professor Schreieder has gained more interesting results in work conducted together with different collaborators and PhD students. “We allow the dimension to be arbitrary in our research, and we try to understand how small could a degree be for that given dimension, where most equations still cannot be parameterized,” he explains. “The technical toolbox for this work comes from algebraic cycles, which is an important sub-field of algebraic geometry. The Hodge conjecture, one of the unsolved Millennium problems, lies within this field.”
This conjecture, and the challenge of understanding algebraic cycles and subvarieties, appears at first sight to be entirely unrelated to the rationality problem. This connection had been drawn by Professor Claire Voisin from Paris in 2015 and has since been generalized and refined. The rationality problem in fact touches on
many different fields of mathematics, and Professor Schreieder says it’s very important to remain open to new ideas when studying it. “I am very much influenced by the wider mathematical community. It is extremely important to communicate with other researchers, read their papers and really understand their work,” he stresses. The fields most closely connected with algebraic geometry are complex geometry, topology and number theory, and many of the techniques Professor Schreieder has been using are derived from these disciplines, yet his work also touches on other branches of mathematics. “I’m based in the complex geometry community, and other researchers in the field work with polytopes and combinatorics for example,” he continues.
The work of French mathematician Alexander Grothendieck was very influential in this respect, helping to establish connections and relationships “ It is possible to parameterize the pythagorean equation, and in general it is always possible to parameterize the solutions to equations in degree 2. However, it gets much more complicated when you look at more complex equations with a larger degree, or with more variables.”
equation minus two; there is a long history of research in this area, and Professor Schreieder is building on earlier findings in his work. “If the degree is larger than the number of variables, roughly speaking, then the problem is relatively easy. In this type of case it is known that you cannot parameterize the solutions,” he says.
“Then in the ‘90s Professor János Kollár showed that most equations of degree at least roughly two-thirds of the dimension
including stereographic projection and certain cohomology theories, which Professor Schreieder uses to study algebraic cycles. “If you look at the zero set, the equation and all the solutions, this gives a certain geometric shape called a variety. An algebraic cycle is, roughly speaking, certain sub-varieties. We study the potential sub-varieties of a certain variety, and look at whether certain techniques can be used to reach findings
RATIONALGIC
Rationality of varieties and algebraic cycles
Project Objectives
The EU-funded “RationAlgic“ project explores classical algebraic geometry problems using modern methods. It focuses on the rationality problem, examining if solutions to multivariable polynomial equations are parametrized by rational functions, and the study of algebraic cycles, which analyzes a variety’s subvarieties. Unramified cohomology is a key tool in this research.
Project Funding
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 948066 (ERC-StG RationAlgic).
Contact Details
Project Coordinator,
Prof. Dr. Stefan Schreieder
Leibniz University Hannover Institute of Algebraic Geometry Welfengarten 1 30167 Hannover
T: +49 511 762 2248
E: schreieder@math.uni-hannover.de W: https://www.iag.uni-hannover.de/en/ schreieder
J. Lange and S. Schreieder, On the rationality problem for hypersurfaces, Preprint 2024, arXiv: 2409.12834 https://arxiv.org/abs/2409.12834
N. Pavic and S. Schreieder,The diagonal of quartic fivefolds, Algebraic Geometry 10 (6) (2023), 754-778.
S. Schreieder, Stably irrational hypersurfaces of small slopes, J. Amer. Math. Soc. 32 (2019), 1171-1199.
Stefan Schreieder is Head of the Institute for Algebraic Geometry at Leibniz University Hannover. He gained his PhD from the University of Bonn in 2015, and has since held several research positions in both Germany and America. He is one of the editors of the Bulletin and the journal of the London Mathematical Society.
between algebraic geometry and other subjects. While his own background is in complex geometry, Professor Schreieder has gained knowledge of other fields and branches of mathematics during the course of his research, helping shape his ideas and the overall direction of the RationAlgic project.“In order to write research papers and develop new approaches I had to learn about certain aspects of number theory, while I could also use some geometric ideas that I was very familiar with,” he outlines. This breadth of interests is reflected in Professor Schreieder’s own research agenda, which extends into other
equations can be parameterized and which can’t. “I would like to improve these degree and dimension bonds further,” he says. This is not something that can be mapped or predicted however, and to an extent progress relies on moments of inspiration, sometimes drawing on ideas and concepts developed by other researchers. “I try to read other people’s work, and also continue with my previous work. At some point key ideas can emerge,” continues Professor Schreieder. “The big improvement of previous results that I achieved in 2019 was maybe not expected at the time, yet it was clear to me that it was significant.”
“ The degree of an equation is, broadly speaking, the highest power term within it – for example, the degree of x3+xy2 would be 3 – while the dimension is the number of variables within the equation minus two.”
areas beyond the rationality problem. “I am also interested in problems related to the Hodge conjecture for example, and other related problems. Some techniques relevant to this work can also be useful for the rationality problem,” he says.
Parameterizing solutions to polynomial equations
This remains the primary focus in the RationAlgic project, which is currently around two-thirds of the way through its six-year funding term. A number of research papers are in the process of being written, with Professor Schreieder exploring new ideas as he seeks to draw a clearer picture of which polynomial
The RationAlgic project team are now aiming to build on this earlier research, and show that if the degree of an equation is sufficiently large compared to the dimension, then it is impossible to parameterize the solutions. At the same time Professor Schreieder and his colleagues are also working on a number of other papers, collaborating and sharing new ideas that could open up further avenues of investigation. “There is a group of people working here at the Institute on a number of fields related to the rationality problem. Our team, including several post-docs and PhD students, works together and influences each other,” says Professor Schreieder.
