Related papers: Combinatorial and inductive methods for the tropic…
This paper is a follow-up of a previous work in which we show that, for a $3$-edge connected tropical curve $\Gamma$, the existence of a divisor of degree $3$ and Baker-Norine rank at least $1$ in $\Gamma$ is equivalent to the existence of…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…
There is a remarkable connection between the maximum clique number and the Lagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. This connection and its extensions were successfully employed in optimization to provide…
We prove (without exceptions) the existence of irredundant tensor decompositions with the number of addenda equal to rank $+1$. We also discuss the existence of decompositions with more than the tensor rank terms, which are concise, while…
In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…
We generalize the method of combinatorial telescoping to the case of multiple summations. We shall demonstrate this idea by giving combinatorial proofs for two identities of Andrews on parity indices of partitions.
We propose an alternating minimization heuristic for regression over the space of tropical rational functions with fixed exponents. The method alternates between fitting the numerator and denominator terms via tropical polynomial…
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to…
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
We begin a systematic study of the enumerative combinatorics of mixed succession rules, which are succession rules such that, in the associated generating tree, the nodes are allowed to produce their sons at several different levels…
We develop a martingale approximation framework yielding quantitative maximal large deviations estimates for invertible dynamical systems. From suitable decay of correlations, we deduce these estimates and, as an application, we obtain…
We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone…
In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative…
An important conjecture in additive combinatorics, number theory, and algebraic geometry posits that the partition rank and analytic rank of tensors are equal up to a constant, over any finite field. We prove the conjecture up to a…
We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made…
The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…
Given a graph $G$, the maximal induced subgraphs problem asks to enumerate all maximal induced subgraphs of $G$ that belong to a certain hereditary graph class. While its optimization version, known as the minimum vertex deletion problem in…
We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the…
In recent work, the authors derived a tropical interpretation of monotone and strictly monotone double Hurwitz numbers. In this paper, we apply the technique of tropical flows to this interpretation in order to provide a new proof of the…