25 research outputs found
Bounds on the number of connected components for tropical prevarieties
For a tropical prevariety in Rn given by a system of k tropical polynomials in n variables with degrees at most d, we prove that its number of connected components is less than k+7n−
Tropical Effective Primary and Dual Nullstellens\"atze
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 and algebraic curves from the computational point of view.
This makes it important to study both mathematical and computational aspects of
tropical polynomials.
In this paper we prove a tropical Nullstellensatz and moreover we show an
effective formulation of this theorem. Nullstellensatz is a natural step in
building algebraic theory of tropical polynomials and its effective version is
relevant for computational aspects of this field.
On our way we establish a simple formulation of min-plus and tropical linear
dualities. We also observe a close connection between tropical and min-plus
polynomial systems
Upper bounds on Betti numbers of tropical prevarieties
We prove upper bounds on the sum of Betti numbers of tropical prevarieties in
dense and sparse settings. In the dense setting the bound is in terms of the
volume of Minkowski sum of Newton polytopes of defining tropical polynomials,
or, alternatively, via the maximal degree of these polynomials. In sparse
setting, the bound involves the number of the monomials.Comment: 9 pages. Upper bounds are slightly improve