Related papers: On a tropical dual Nullstellensatz
Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties…
Grigoriev and Podolskii (2018) have established a tropical analogue of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay…
Tropical algebra emerges in many fields of mathematics such as algebraic geometry, mathematical physics and combinatorial optimization. In part, its importance is related to the fact that it makes various parameters of mathematical objects…
This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra,…
Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients.…
We show factorization of polynomials in one variable over the tropical semiring is in general NP-complete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization…
We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these…
We suggest a version of Nullstellensatz over the tropical semiring, the real numbers equipped with operations of maximum and addition.
We fix the supports A=(A_1,...,A_k) of a list of tropical polynomials and define the tropical resultant TR(A) to be the set of choices of coefficients such that the tropical polynomials have a common solution. We prove that TR(A) is the…
The problem of solving tropical linear systems, a natural problem of tropical mathematics, has already proven to be very interesting from the algorithmic point of view: it is known to be in $NP\cap coNP$ but no polynomial time algorithm is…
We describe a new approach based on tropical optimization techniques to solve the problem of rating alternatives from pairwise comparison data. The problem is formulated to approximate, in the log-Chebyshev sense, pairwise comparison…
We present simple compact proofs of the strong and weak duality theorems of tropical linear programming. It follows that there is no duality gap for a pair of tropical primal-dual problems. This result together with known properties of…
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…
For a tropical univariate polynomial $f$ we define its tropical Hilbert function as the dimension of a tropical linear prevariety of solutions of the tropical Macauley matrix of the polynomial up to a (growing) degree. We show that the…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
In this paper, we review the eigenpair problem in the context of tropical algebra. An important fact that has been largely overlooked in spectral theory of tropical algebra is that the tropical algebraic eigenvalues, which are obtained from…
We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets…
We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.
This thesis delves into the geometry of abstract tropical curves, exploring their complete linear system and associated tropical submodules. We establish a lower bound on the dimension of tropical submodules in terms of the Baker-Norine…
The tropical convex hull of a finite set of points in tropical projective space has a natural structure of a cellular free resolution. Therefore, methods from computational commutative algebra can be used to compute tropical convex hulls.…