Since a tropical Nullstellensatz fails even for tropical univariate polynomials we study a conjecture on a tropical {it dual} Nullstellensatz for tropical polynomial systems in terms of solvability of a tropical linear system with the Cayley matrix associated to the tropical polynomial system. The conjecture on a tropical effective dual Nullstellensatz is proved for tropical univariate polynomials.
We prove a version of a Nullstellensatz for partial exponential fields $(K,E)$, even though the ring of exponential polynomials $K[X_1,ldots,X_n]^E$ is not a Hilbert ring. We show that under certain natural conditions one can embed an ideal of $K[X_1,ldots,X_n]^E$ into an exponential ideal. In case the ideal consists of exponential polynomials with one iteration of the exponential function, we show that these conditions can be met. We apply our results to the case of ordered exponential fields.
In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph $G$, we define the invariant $eta(G)$ as the maximum size of a clique disjoint edge set in $G$. We show that the regularity of the binomial edge ideal of $G$ is bounded above by $eta(G)$. This, in particular, settles a conjecture on the regularity of binomial edge ideals in full generality.
This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.
The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +), Izhakian-Rowens extended and supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. This allows us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.
The goal of this article is to classify unramified covers of a fixed tropical base curve $Gamma$ with an action of a finite abelian group G that preserves and acts transitively on the fibers of the cover. We introduce the notion of dilated cohomology groups for a tropical curve $Gamma$, which generalize simplicial cohomology groups of $Gamma$ with coefficients in G by allowing nontrivial stabilizers at vertices and edges. We show that G-covers of $Gamma$ with a given collection of stabilizers are in natural bijection with the elements of the corresponding first dilated cohomology group of $Gamma$.