We introduce a simple, easy to implement, and computationally efficient tropical convolutional neural network architecture that is robust against adversarial attacks. We exploit the tropical nature of piece-wise linear neural networks by embedding the data in the tropical projective torus. This can be accomplished with a single additional hidden layer called a tropical embedding layer, and can in principle be added to any neural network architecture. We study the geometry of the resulting decision boundary, and find that like adversarial training and various regularization techniques that have been proposed, adding the tropical embedding layer tends to increase the number of linear regions associated with the decision boundaries. Our numerical experiments show that our approach achieves state-of-the-art levels of adversarial robustness, while requiring much less computational time than adversarial training.

Abstract The dynamics of open quantum systems described by the Lindblad master equation follows according to non-Hermitian operators. As a result, such systems can host non-Hermitian degeneracies called Liouvillian exceptional points (EPs). In this work, we show that Newton polygons and tropical geometric approach allow identification and characterization of Liouvillian EPs. We use two models -- dissipative spin 1/2 system and dissipative superconducting qubit system -- to illustrate our method. We demonstrate that our approach captures the anisotropy and order of the Liouvillian EPs, while also revealing the subtle dependence on the form of the perturbation. Our analytical analysis is supplemented by direct numerical calculations of the scaling and exchange of eigenvalues around Liouvillian EPs. Our analytical approach could be useful in understanding and designing Liouvillian EPs of desired order.

Coulomb branch amplitudes of maximally supersymmetric Yang-Mills theory display infrared properties different from their conformal counterparts. While the four-leg amplitude is known to very high perturbative orders, amplitudes of higher multiplicity fall into an uncharted territory starting already from two loops. The reason for this is that they are not easily amenable to traditional techniques like canonical differential equations due to the uncontrolled swelling of solutions to integration-by-parts identities. In this paper, we break the barrier for the five-leg amplitude using a technique based on the analysis of Newton polytopes corresponding to Feynman/Schwinger integrands and their tropical geometry. Specifically, we analytically evaluate the near mass-shell limit of the off-shell pentabox in terms of Goncharov polylogarithms.

Abstract The main goal of this paper is to establish the higher dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions (tropical holomorphic maps) in several real variables, such as the proximity function, counting function and characteristic function, the first main theorem, higher dimensional tropical versions of the logarithmic derivative lemmas. Based on this, for algebraically non‐degenerate tropical holomorphic maps with subnormal growth from into tropical projective space intersecting tropical hypersurfaces with degree , we then obtain the second main theorem where and . Our work in this paper significantly generalizes previous results by Korhonen–Tohge [Adv. Math. 298 (2016), 693–725] and Cao–Zheng [Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 (2025), no. 4, 2105–2144].

Abstract This paper generalises the homeomorphism theorem behind Viro’s combinatorial patchworking of hypersurfaces in toric varieties to arbitrary codimension using tropical geometry. We first define the patchwork of a polyhedral space equipped with a real phase structure. When the polyhedral subspace is tropically non-singular, we show that the patchwork is a topological manifold. When a non-singular tropical variety appears as a tropical limit of a real analytic family, we show that the real part of a fibre of the family near the tropical limit is homeomorphic to the patchwork. Finally, we extend the spectral sequence introduced by the last two authors in the case of hypersurfaces to non-singular tropical varieties with real phase structures. As a corollary, we obtain bounds on the Betti numbers of the patchwork in terms of the dimensions of the tropical homology groups with coefficients modulo two.

This paper focuses on studying the configuration spaces of graphs realised in C 2 \mathbb C^2 , such that the configuration space is, after normalisation, one dimensional. If this is the case, then the configuration space is, generically, a smooth complex curve, and can be seen as a Riemann surface. The property of interest in this paper is the genus of this curve. Using tropical geometry, we give an algorithm to compute this genus. We provide an implementation in Python and give various examples.

Hurwitz numbers enumerate branched morphisms between Riemann surfaces. For a fixed elliptic target, Hurwitz numbers are intimately related to mirror symmetry following work of Dijkgraaf. In recent work of Chapuy and Dołęga, a new variant of Hurwitz numbers with fixed genus 0 target was introduced that includes maps between non-orientable surfaces. These numbers are called b -Hurwitz numbers and are polynomials in a parameter b which measures the non-orientability of the involved maps. An interpretation in terms of factorisations of b -Hurwitz numbers for b=1 , so-called twisted Hurwitz numbers, was found in work of Burman and Fesler. In previous work, the authors derived a tropical geometry interpretation of these numbers. In this paper, we introduce a natural generalisation of twisted Hurwitz numbers with elliptic targets within the framework of symmetric groups. We derive a tropical interpretation of these invariants, relate them to Feynman integrals and derive an expression as a matrix element of an operator in the bosonic Fock space.

We present a new approach for proving that, for a given n, there are finitely many equivalence classes of planar central configurations in the Newtonian n-body problem for generic masses. The human part of the proof relies on tropical geometry. The crux of our technique is in a computation that we have completed for n<=5, thus confirming the celebrated result of Albouy and Kaloshin.

Ceballos and Pons introduced the s \mathrm {s} -weak order on s \mathrm {s} -decreasing trees, for any weak composition s \mathrm {s} . They proved that it has a lattice structure and further conjectured that it can be realized as the 1 1 -skeleton of a polyhedral subdivision of a polytope. We answer their conjecture in the case where s \mathrm {s} is a strict composition by providing three geometric realizations of the s \mathrm {s} -permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second one, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third one, obtained using tropical geometry, is the 1 1 -skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured.

In 2015, G. Mikhalkin introduced a refined count for real rational curves in a toric surface which pass through some points on the toric boundary of the surface. The refinement is provided by the value of a so-called quantum index. Moreover, he proved that the result of this refined count does not depend on the choice of the points. The correspondence theorem allows one to compute these invariants using the tropical geometry approach and the refined Block–Göttsche multiplicities. In this paper we give a recursive formula for these invariants, that leads to an algorithm to compute them.

We define arroids as an abstract axiom set encoding the intersection properties of arrangements of curves. The tropicalization of the complement of an arrangement of curves meeting pairwise transversely is shown to be determined by the associated arroid. We give conditions for when the cohomology of the complement of an arrangement is computable using tropical cohomology, and we give criteria for when the complement is a maximal variety in terms of tropical geometry.

We say that a tropical subvariety V R n is B-realizable if it can be lifted to an analytic subset of ./ n .When V is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift L V .C / n .We prove that whenever L V has well-defined Floer cohomology, we can find for each point of V a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with L V is nonvanishing.Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever L V is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety V is B-realizable.As an application, we show that the Lagrangian lift of a genus-0 tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-0 tropical curves are B-realizable.We also prove that tropical curves inside tropical abelian surfaces are B-realizable.

Abstract We describe an application of tropical moduli spaces to complex dynamics. A post‐critically finite branched covering of induces a pullback map on the Teichmüller space of complex structures of ; this descends to an algebraic correspondence on the moduli space of point‐configurations of . We make a case for studying the action of the tropical moduli space correspondence by making explicit the connections between objects that have come up in one guise in tropical geometry and in another guise in complex dynamics. For example, a Thurston obstruction for corresponds to a ray that is fixed by the tropical moduli space correspondence, and scaled by a factor . This article is intended to be accessible to algebraic and tropical geometers as well as to complex dynamicists.

Tropical Principal Component Analysis (PCA) is an analogue of the classical PCA in the setting of tropical geometry, and applied it to visualize a set of gene trees over a space of phylogenetic trees, which is a union of lower-dimensional polyhedral cones in an Euclidean space with dimension m(m−1)/2, where m is the number of leaves. In this paper, we introduce a projected gradient descent method to estimate the tropical principal polytope over the space of phylogenetic trees, and we apply it to an Apicomplexa dataset. With computational experiments against Markov Chain Monte Carlo (MCMC) samplers, we show that our projected gradient descent method yields a lower sum of tropical distances between observations and their projections onto the estimated best-fit tropical polytope, compared with the MCMC-based approach.

Toric Vector Bundles, Valuations and Tropical Geometry

Abstract We use tropical and nonarchimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space . In particular, we are interested in the choices of parameters for which the generic root count is attained. Our families are given as subschemes where is a relative torus over . We generalize Bernstein's theorem from an intersecting family of hypersurfaces to an intersecting family of higher‐codimensional schemes , replacing the mixed volume by a tropical intersection product. Central to our work is the notion of tropical flatness of around a point , which allows us to transfer tropical properties of the fiber over to generic properties. We show that tropical flatness holds over a dense open subset of the Berkovich analytification , and that the tropical intersection number is attained as a root count at all around which the ’s are tropically flat and the tropical prevariety of the fibers is bounded. We then study the generic root count of a wide class of parametrized square polynomial systems. This, in particular, gives tropical formulas for the volumes of Newton–Okounkov bodies, and the number of complex steady states of chemical reaction networks.

The CEGM formalism offers a general framework for scattering amplitudes, which rests on Grassmannians, moduli spaces and tropical geometry. The physical implications of this generalization are still to be understood. Conventional wisdom says that key features of scattering amplitudes, like factorization at their poles into lower-point amplitudes, are associated to their singularities. The factorization behavior of CEGM amplitudes at their poles is interesting but complicated. Recent developments have revealed important properties of standard particle and string scattering amplitudes from factorizations, known as splits, that happen away from poles. In this paper we introduce a kinematic subspace on which the CEGM amplitude splits into very simple rational functions. These functions, called simplex amplitudes, arise from stringy integrals for the multivariate beta function, and also from restricting the biadjoint scalar amplitude in quantum field theory to certain kinematic loci. Using split kinematics we also discover a specific class of zeros of the CEGM amplitude. Our construction rests on viewing positive moduli space as a product of simplices, and it suggests a novel approach for deriving scattering amplitudes from tropical determinantal varieties.

We study Higgs branches of field theories with 8 supercharges in 5 and 6 dimensions, focusing on theories realised on 5-brane webs in Type IIB with an O7+ plane, or a D6-D8-NS5 brane system in Type IIA in the presence of an O8+ plane. We find magnetic quivers for the Higgs branches of these theories. The main consequence of the presence of the orientifold is that it renders the magnetic quiver to be non-simply-laced. We propose a contribution of the O7+ to the usual stable intersection number of 5-branes from tropical geometry, and show that it is consistent with Fayet-Iliopoulos deformations of magnetic quivers which represent mass deformations of 5d SQFTs. From the magnetic quivers, we compute phase diagrams and highest weight generating functions for the Higgs branches, enabling us to identify the global form of the flavour symmetry for several families of 5d SQFTs; among them Bhardwaj’s rank-1 theory. For 6d theories realised on a −4 curve, we observe the appearance of an additional D4 slice on top of the phase diagram as one goes to the tensionless limit.

Abstract Using tropical geometry, one can translate problems in enumerative geometry to combinatorial problems. Thus, tropical geometry is a powerful tool in enumerative geometry over the complex and real numbers. Results from A 1 \mathbb{A}^{1} -homotopy theory allow to enrich classical enumerative geometry questions and get answers over an arbitrary field. In the resulting area, A 1 \mathbb{A}^{1} -enumerative geometry, the answer to these questions lives in the Grothendieck–Witt ring of the base field 𝑘. In this paper, we use tropical methods in this enriched set up by showing Bézout’s theorem and a generalization, namely the Bernstein–Kushnirenko theorem, for tropical hypersurfaces enriched in GW ⁡ ( k ) \operatorname{GW}(k) .

The main aim of this paper is to show the interconnections between Łukasiewicz logic and algebraic geometry using algebraic, geometric and logical instruments. We continue our investigation into a new algebraic geometry based on idempotent semifields, in particular those related to MV-algebras, describing categorical equivalence between different structures related to tropical geometry and many valued logics. Further, we describe such connections in terms of topoi.