## Tropical Differential Algebra Workshop

### December 3–7, 2019

### Queen Mary University of London

Tuesday's participants

### Structure of Workshop:

The themes of the Tropical Differential Algebra Workshop are tropical geometry, differential algebra and their interactions. Tuesday, the first day of the workshop, will consist of mini-courses on these three themes; we particularly recommend attending this day for students. All day Wednesday and the mornings on Thursday and Friday will be dedicated to talks, including hour-long talks from experts and half-hour contributed talks from students. The remainder of the workshop time will be dedicated to research, together or in a few groups.

### Organizers:

### Speakers

Youren Hu

Stefano Mereta

François OLLIVIER by Skype

Fernando Zhapa

Tuesday's mini-courses will be delivered by François Boulier and Jeffrey Giansiracusa. François has provided notes for his course (these are the final version, newer than the draft that was circulated).

### Registration:

If you are interested in attending the workshop, please register here. The deadline to apply for financial support has now passed.

### Conference dinner:

A conference dinner will be held at Verdi's, along Mile End Road just to the west of campus (walking directions), starting at **18:30 on Wednesday**. Here is the set menu for the dinner.

### Accommodation:

For hotel suggestions, see this list of accommodation options from a recent Queen Mary-hosted conference. One of the organisers has also found the Plaza Hotel London to be fair and affordable.

### How to get to the conference venues:

The conference will take place on the Mile End campus (map here) of Queen Mary University of London, to which the nearest Underground stations are Mile End and Stepney Green. Tuesday and Wednesday's events are in the GO Jones building, #25 on the map (we begin in room 610, on the 6th floor). From Thursday on we are in the Maths building, #4 on the map.

See the Transport for London website for Underground travel directions.

## Titles and Abstracts of Talks:

### Fuensanta Aroca: NEWTON’S LEMMA FOR DIFFERENTIAL EQUATIONS

(Joint work with Giovanna Ilardy)

Abstract: The Newton method for plane algebraic curves is based on the following remark: the first term of a series, root of a polynomial with coefficients in the ring of series in one variable, is a solution of an initial equation that can be determined by the Newton polygon. Given a monomial ordering in the ring of polynomials in several variables, we describe the systems of initial equations that satisfy the first terms of the solutions of a system of partial differential equations. As a consequence, we extend Mora and Robbiano’s Groebner fan to differential ideals.

### Cristhian Garay: Differential semirings

Abstract: We discuss the concept of derivations defined on a (commutative) semiring, and on modules over them. This serves as a generalization of the concepts of differential rings, and idempotent differential rings, of which tropical differential rings are a particular case.

The aim is that this analogues of the classical differential objects will help the study of geometric objects defined over semirings, like tropical (differential) algebraic varieties.

### Jeffrey Giansiracusa: Tropicalization, universal tropicalization, and analytification

Abstract: The tropicalization of a scheme depends on the choice of an embedding into an appropriate ambient space (usually a toric variety). The analytification is intrinsic, and it has a canonical map onto each tropicalization. Building on ideas from Kontsevich, Payne showed that the Berkovich analytification of a quasi-projective scheme is homeomorphic to the limit of the tropicalizations. Thus we think of the analytification as an intrinsic/universal tropicalization. In this talk I will show how to make this heuristic idea a reality. I will explain how thinking about semirings, bend relations, and the moduli space of valuations, leads to 1) a description of the analytification as a genuine tropicalization with respect to a certain universal embedding, and 2) a tropical-scheme refinement of Payne's limit theorem. Since Berkovich theory provides a compelling foundational language for non-archimedean differential equations, I hope that the tropical understanding of Berkovich analytification will play a role in developing a broader theory of tropical differential equations.

### Dmitry Grigoryev: TROPICAL RECURRENT SEQUENCES AND TROPICAL ENTROPY

Abstract : Tropical recurrent sequences satisfying a given real vector, is a tropical analog of linear recurrent sequences. In addition, we consider tropical minimal recurrent sequences and study the question, for which vectors all troipical tropical minimal recurrent sequences are periodic in case when the Newton polygon of the vector has a single bounded edge. We introduce a tropical (resp. tropical minimal) entropy as a measure of the space of tropical (resp. minimal) recurrent sequences and prove that the tropical entropy vanishes iff the Newton polygon is regular and all the points being the vertices of the polygon. If the tropical entropy is positive it is greater or equal to 1/6. We discuss the conjecture that the tropical minimal entropy vanishes iff all the tropical minimal recurrent sequences are periodic. We consider a miltidimensional extension of the tropical entropy and the open problems.

### Youren Hu: Tropical Differential Groebner Bases

Abstract : The concept of tropical differential Groebner basis is introduced, which is a natural generalization of the tropical Groebner basis to the recently introduced tropical differential algebra. Like the differential Groebner basis, the tropical differential Groebner basis generally contains an infinite number of elements. We give a Buchberger style criterion for the tropica differential Groebner basis. For differential ideals generated by homogeneous linear differential polynomials with constant coefficients, we give a complete algorithm to compute the finite tropical differential Groebner basis.

### Stefano Mereta: Scheme theoretic approach to differential tropical geometry

Abstract : Starting from the presentation of tropical differential equations as in the work of Grigoriev and the following paper by Aroca, Toghani and Garay, in which the fundamental theorem of differential tropical geometry has been proven, I’ll try to put their work in the framework of semiring theory. I’ll go through the definitions of tropical Leibniz rule to define a differential scheme over a semiring whose points are the solutions of a differential tropical equation. The long term goal will be to define a universal differential semiring whose spectrum is the moduli space of what we call differential enhancements of a given valuation.

### François Ollivier: Jacobi's bound and tropical determinant

Abstract : Around 1840, C.G.J. Jacobi wrote an algorithm to compute what is now known as the "tropical determinant", using some pioneering tools related to graph theory and shortest paths problems. His aim was to compute a bound on the order of a differential system. Jacobi's bound, still conjectural in the general case, claims that the order of a component of a system P is the tropical determinant of the matrix A=(a_{i,j}), with a_{i,j}:=ord_{x_j} P_i. The basis idea to compute this bound is to compute the minimal canon of A, that is the minimal matrix a_{i,j}+ \ell_i, obtained by increasing some rows so that for some permutation \sigma a_{i,\sigma(i)} is maximal in its column. Jacobi's bound is reached iff some "truncated determinant" is non-zero: then, the minimal integers \ell_i are such that one may compute a characteristic set of the component, by differentiating equation P_i at most \ell_i times. If one denotes by \cO the tropical determinant of A and by \cO_{i,j} the tropical determinant of A deprived of line i and row j, we have the obvious formula: \cO=\max_{j=1}^n \cO_{i,j}+a_{i,j}. It was used by Nanson and Jordan in two heuristic "proofs" of Jacobi's bound, together with the tropical formula :

$$ \sum_{i=1}^{n}\left(\cO_{i,j_{0}}+1\right)=1+\sum_{j\ne j_{0}}\left(\max_{i=1}^{n}(\cO_{i,j_{0}}+a_{i,j})+1\right)$$

Jacobi's algorithm is similar to Kuhn's "Hungarian method" using the minimal covers of Jenő Egerváry. With some improvements, the complexity of both algorithms is O(n^3). One may achieve O(n^(5/2)) with integers coefficients. One trouble when computing the tropical determinant does not behave well with the tropical matrix product, unless special cases, mostly when matrices are canons or transposes of canons. For the best of my knowledge, the question of minimal bounds of complexity remain open, in the case of integer coefficient as well as for general commutative ordered groups. With arbitrary monoids, an exponential complexity can be met.

### Yue Ren: Computational tropical geometry - algorithms, applications, and software

Abstract: This talk is a brief and introductory survey on algorithms and software for tropical geometry, and their applications. We provide a

short overview of the software systems Gfan, Polymake, and Singular, with emphasis on their present and future features as well as their

usage. Moreover, in light of very recent works on Gröbner bases of tropical differential ideals and the Fundamental Theorem of Tropical

Differential Geometry, we highlight the importance of their classical counterparts for computational tropical geometry.

### Ben Smith: Higher rank tropical hypersurfaces

Abstract: Tropical hypersurfaces arising from polynomials over the Puiseux series are well studied and understood objects. The picture becomes less clear when considering Puiseux series in multiple indeterminates. Unlike their rank one counterparts, these higher rank tropical hypersurfaces are not ordinary polyhedral complexes but still have a large amount of structure. We shall consider the various polyhedral-like structures we may place on these objects, and the topological issues that arise in transitioning to tropical geometry of higher rank.

### Mercedes Haiech: The arc scheme from the point of view of differential algebra

Abstract: Given a variety (or a scheme) X defined over a field k, one can associated with it a geometrical object called the arc scheme of X which is, roughly speaking, the formal germs of curves living on the variety. This object was first introduced in the sixties and its study can give informations about the singularities of the variety. In this talk we will show a classical interpretation of the arc scheme in terms of differential algebra.

### Fernando Zhapa: Development of a Tropical Algebraic Geometry package in the Haskell programming language

Abstract: In this work, we present a package of Tropical Geometry written in the Haskell programming language. We use the advantages of functional programming such as dependent-typing and composability to implement polynomial behaviour and the corresponding functions to perform the computation of its tropical hypersurface and the graph of its curve. At this point, the package allows computing varieties in R2 by performing the intersection of tropical hypersurfaces generated by the polynomials by a given ideal. One of the primary purposes of this package was to compute the tropical variety (or pre-variety) of an ideal. However, the data types and functions used in the process are also considered as part of the package and include the implementation of tropical numbers, tropical matrices, and the functions related to polyhedral geometry and computational geometry. Moreover, another feature is the arbitrary precision since Haskell implements arbitrary precision integers natively, therefore we offer exact computations in the process.

### Contact:

If you have any question please send email to the organizers at z.toghani@qmul.ac.uk and a,fink@qmul.ac.uk.

### The workshop is supported by:

the European Commission, having received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement number 792432

the School of Mathematical Sciences at Queen Mary University of London.