Related papers: Tropical rational equivalence on R^r

We study rational functions over finite fields under PGL-equivalence. We say that $f, g \in \Bbb F_q(X)$ are \emph{equivalent} if there exist $\psi, \phi \in \Bbb F_q(X)$ of degree one such that $g = \psi \circ f \circ \phi$. Most…

We prove a theorem on the intersection theory over a Noetherian local ring $R$, which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and…

We investigate rational maps with period one and two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is $d$ and the cluster is fixed, the Thurston class of a rational map is fixed by the…

The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…

A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…

Let $k$ be the function field of a complex curve or the field $C((t))$. We show that for a smooth complete intersection $X$ of $r$ hypersurfaces in $P^n_k$ of respective degrees $d_1,...,d_r$ with $\sum d_i^2\leq n+1$ the R-equivalence on…

Any map of schemes $X\to Y$ defines an equivalence relation $R=X\times_Y X\to X\times X$, the relation of "being in the same fiber". We have shown elsewhere that not every equivalence relation has this form, even if it is assumed to be…

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.…

Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.

We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts…

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the…

We define arroids as an abstract axiom set encoding the intersection properties of arrangements of curves. The tropicalization of the complement of arrangement of curves meeting pairwise transversely is shown to be determined by the…

We consider toric maximum likelihood estimation over the field of Puiseux series and study critical points of the likelihood function using tropical methods. This problem translates to finding the intersection points of a tropical affine…

It has been proposed that the number of tropical cyclones as a function of the energy they release is a decreasing power-law function, up to a characteristic energy cutoff determined by the spatial size of the ocean basin in which the storm…

We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove…

Let $A$ be a rational function. For any decomposition of $A$ into a composition of rational functions $A=U\circ V$ the rational function $\widetilde A=V\circ U$ is called an elementary transformation of $A$, and rational functions $A$ and…

We prove that every connected component of an intersection of tropical hypersurfaces contains a point of their stable intersection unless their stable intersection is empty. This is done by studying algebraic hypersurfaces that tropicalize…

We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral…

We target the problem of provably computing the equivalence between two complex expression trees. To this end, we formalize the problem of equivalence between two such programs as finding a set of semantics-preserving rewrite rules from one…

We consider a reinforcement learning framework where agents have to navigate from start states to goal states. We prove convergence of a cycle-detection learning algorithm on a class of tasks that we call reducible. Reducible tasks have an…