# Computational tropical geometry and its applications

Lead Research Organisation:
Swansea University

Department Name: College of Science

### Abstract

Tropical geometry is a young area of mathematics which studies combinatorial objects arising from polynomial equations. These so-called tropical varieties arise naturally in many areas of mathematics and beyond, such as phylogenetics in biology, celestial mechanics in physics, and auction theory in economics. Wherever they arise, tropical varieties often allow new computational approaches to existing problems. In the UK, the Bank of England has been using tropical geometry since the financial crises to allocate money to the UK financial system. In France, tropical geometry is used for optimisation of load balancing of mobile networks, and performance analysis of emergency call centres.

This research projects aims at establishing tropical geometry as a powerful and versatile tool for computational questions in applied sciences and industry beyond optimisation. To this end, we pursue concrete applications as well as improvements of computational methods. The final deliverable is a comprehensive open source software system for tropical algebraic geometry with strong emphasis on its wide spectrum of applications. We focus on three main problems, which were chosen to maximise the impact and the range of techniques that they encompass.

The first problem revolves around systems of polynomial equations, which are ubiquitous in applied science. They describe the steady states of chemical reaction networks, the range of movement of a robot arm, or the binding behaviour of ligands in a biochemical system. For over two decades, the state of the art for solving such systems has been homotopy continuation, which works by carefully deforming an easy start system to the target system while tracing all solutions along the way.

We seek to improve the existing capabilities, in particular for the type of polynomial systems which arise in the aforementioned applications. While ideas to apply tropical geometry to homotopy continuation have already been studied, all past approaches have failed due to questions of efficiency. However, the last couple of years have seen significant algorithmic breakthroughs in tropical geometry, which we will exploit and build upon.

The second problem involves p-adic numbers, which are an indispensable class of fields for number theory. This not only makes them important for the applications of tropical geometry in number theory, but also entails a vast array of number theoretic tools available exclusively over them. Hence a good grasp on tropical geometry over p-adics numbers is an imperative for both theory and practice.

That being said, computationally, tropical geometry over p-adic numbers has been neglected due to the unique algorithmic challenges they pose. We seek to remedy this situation and explore computational aspects of tropical geometry specifically over p-adic numbers, facilitated by recent trends in computer algebra.

The third problem involves Gröbner bases, which have long history in computational algebraic geometry and adjacent fields such as cryptography. Furthermore, the past decade featured an explosion of algebro-geometric techniques in areas outside of mathematics. As such, Gröbner bases have gained traction both as tool for studying polynomial systems and as object of interest themselves, e.g., as Markov bases in algebraic statistics. However, Gröbner bases are notoriously hard to compute, which severely inhibits their use in practical applications.

We will investigate so-called saturating Gröbner bases. In general, polynomial unknowns represent arbitrary elements of the coefficient field, and all operations within a Gröbner basis computation respect this ambiguity. In practice, one is often only interested in specific solutions, e.g. strictly positive real solutions. Saturating Gröbner basis algorithm are symbolic algorithms which are capable of exploiting this numerical information that is abundant in many applications and use it to speed up its performance.

This research projects aims at establishing tropical geometry as a powerful and versatile tool for computational questions in applied sciences and industry beyond optimisation. To this end, we pursue concrete applications as well as improvements of computational methods. The final deliverable is a comprehensive open source software system for tropical algebraic geometry with strong emphasis on its wide spectrum of applications. We focus on three main problems, which were chosen to maximise the impact and the range of techniques that they encompass.

The first problem revolves around systems of polynomial equations, which are ubiquitous in applied science. They describe the steady states of chemical reaction networks, the range of movement of a robot arm, or the binding behaviour of ligands in a biochemical system. For over two decades, the state of the art for solving such systems has been homotopy continuation, which works by carefully deforming an easy start system to the target system while tracing all solutions along the way.

We seek to improve the existing capabilities, in particular for the type of polynomial systems which arise in the aforementioned applications. While ideas to apply tropical geometry to homotopy continuation have already been studied, all past approaches have failed due to questions of efficiency. However, the last couple of years have seen significant algorithmic breakthroughs in tropical geometry, which we will exploit and build upon.

The second problem involves p-adic numbers, which are an indispensable class of fields for number theory. This not only makes them important for the applications of tropical geometry in number theory, but also entails a vast array of number theoretic tools available exclusively over them. Hence a good grasp on tropical geometry over p-adics numbers is an imperative for both theory and practice.

That being said, computationally, tropical geometry over p-adic numbers has been neglected due to the unique algorithmic challenges they pose. We seek to remedy this situation and explore computational aspects of tropical geometry specifically over p-adic numbers, facilitated by recent trends in computer algebra.

The third problem involves Gröbner bases, which have long history in computational algebraic geometry and adjacent fields such as cryptography. Furthermore, the past decade featured an explosion of algebro-geometric techniques in areas outside of mathematics. As such, Gröbner bases have gained traction both as tool for studying polynomial systems and as object of interest themselves, e.g., as Markov bases in algebraic statistics. However, Gröbner bases are notoriously hard to compute, which severely inhibits their use in practical applications.

We will investigate so-called saturating Gröbner bases. In general, polynomial unknowns represent arbitrary elements of the coefficient field, and all operations within a Gröbner basis computation respect this ambiguity. In practice, one is often only interested in specific solutions, e.g. strictly positive real solutions. Saturating Gröbner basis algorithm are symbolic algorithms which are capable of exploiting this numerical information that is abundant in many applications and use it to speed up its performance.

### Planned Impact

Research in pure mathematics has had enormous cultural and economic impact in the UK and throughout Europe through contributions to the sciences, engineering, finance, and medicine. Recent contributions of tropical geometry in particular include the Bank of England providing liquidity to the UK financial system since the last financial crisis, Orange optimising load balancing of its mobile networks, and the Paris police and fire departments analysing the performance of their emergency call centres. Furthermore, researchers at the University of Birmingham are currently working on applying tropical geometry to the optimisation of the UK train schedules.

In addition to general improvements to the toolbox of tropical geometry that is being used in the aforementioned examples, the proposed research programme directly targets a problem that is ubiquitous in applied science: solving systems of polynomial equations, in particular those which arise specifically from chemical reaction networks and in mechanical design.

Furthermore, the methods that I will develop for tropical geometry have several natural applications in the context of applied algebraic geometry outside of mathematics. These include:

- Biology (Phylogenetics)

Tree spaces in tropical geometry admit a natural structure of tropical varieties, whose computation is one of the central tasks in the programme.

- Biology (Networks) and statistics

Biological networks and certain statistical models can be modelled using polynomial equations, for whom a Gröbner basis is of particular interest. For example, for toric models they are related to the notion of Markov bases.

One of my major aims with this research programme is to provide new opportunities for collaboration between researchers in pure mathematics and applied sciences, and improve the sustainability of open source mathematical software by creating new avenues for industrial funding.

Moreover, my fellowship includes a postgraduate researcher, who will bnefit directly because it will give them experiences of collaborative interdisciplinary projects and enable them to identify, develop and also learn new transferable skills, such as self-awareness, initiative, teamwork, communication, networking, problem solving, flexibility, etc. These transferable skills will be essential for their future career.

Finally, there are several educational and outreach activities in which I will engage during the fellowship with the support of the Further Maths Support Programme Wales, the Oriel Science centre and the Swansea Science Festival which are all hosted in Swansea. These activities include talks targeted towards high school students, and exhibitions targeted towards all ages.

In addition to general improvements to the toolbox of tropical geometry that is being used in the aforementioned examples, the proposed research programme directly targets a problem that is ubiquitous in applied science: solving systems of polynomial equations, in particular those which arise specifically from chemical reaction networks and in mechanical design.

Furthermore, the methods that I will develop for tropical geometry have several natural applications in the context of applied algebraic geometry outside of mathematics. These include:

- Biology (Phylogenetics)

Tree spaces in tropical geometry admit a natural structure of tropical varieties, whose computation is one of the central tasks in the programme.

- Biology (Networks) and statistics

Biological networks and certain statistical models can be modelled using polynomial equations, for whom a Gröbner basis is of particular interest. For example, for toric models they are related to the notion of Markov bases.

One of my major aims with this research programme is to provide new opportunities for collaboration between researchers in pure mathematics and applied sciences, and improve the sustainability of open source mathematical software by creating new avenues for industrial funding.

Moreover, my fellowship includes a postgraduate researcher, who will bnefit directly because it will give them experiences of collaborative interdisciplinary projects and enable them to identify, develop and also learn new transferable skills, such as self-awareness, initiative, teamwork, communication, networking, problem solving, flexibility, etc. These transferable skills will be essential for their future career.

Finally, there are several educational and outreach activities in which I will engage during the fellowship with the support of the Further Maths Support Programme Wales, the Oriel Science centre and the Swansea Science Festival which are all hosted in Swansea. These activities include talks targeted towards high school students, and exhibitions targeted towards all ages.

## People |
## ORCID iD |

Yue Ren (Principal Investigator / Fellow) |