Related papers: Matrix-F5 algorithms and tropical Gr\"obner bases …
Tropical semiring has proven successful in several research areas, including optimal control, bioinformatics, discrete event systems, or solving a decision problem. In previous studies, a matrix two-factorization algorithm based on the…
We present algorithms for computing zero-dimensional tropical varieties as implemented in OscarZerodimensionalTropicalization.jl. The algorithms include a mathematical workaround for a common practical issue arising when working with…
Using evaluation at appropriately chosen points, we propose a Gr\"obner basis free approach for calculating the secondary invariants of a finite permutation group. This approach allows for exploiting the symmetries to confine the…
In this paper, we focus on computing the kernel of a map of polynomial rings $\varphi$. This core problem in symbolic computation is known as implicitization. While there are extremely effective Gr\"obner basis methods used to solve this…
We prove that there exist uniform $(+,\times,/)$-circuits of size $O(n^3)$ to compute the basis generating polynomial of regular matroids on $n$ elements. By tropicalization, this implies that there exist uniform $(\max,+,-)$-circuits and…
Modular algorithm are widely used in computer algebra systems (CAS), for example to compute efficiently the gcd of multivariate polynomials. It is known to work to compute Groebner basis over $\Q$, but it does not seem to be popular among…
We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of…
We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…
Over the past decade, the Gr\"obner basis theory and automatic solver generation have lead to a large number of solutions to geometric vision problems. In practically all cases, the derived solvers apply a fixed elimination template to…
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
State space subspace algorithms for input-output systems have been widely applied but also have a reasonably well-developedasymptotic theory dealing with consistency. However, guaranteeing the stability of the estimated system matrix is a…
We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the…
A generalized criterion for signature related algorithms to compute Gr\"obner basis is proposed in this paper. Signature related algorithms are a popular kind of algorithms for computing Gr\"obner basis, including the famous F5 algorithm,…
In this article we introduce the recently developed polymake extension TropicalQuarticCurves and its associated database entry in polyDB dealing with smooth tropical quartic curves. We report on algorithms implemented to analyze tropical…
We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity…
In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally…
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to…
The efficiency of Gr\"obner basis computation, the standard engine for solving systems of polynomial equations, depends on the choice of monomial ordering. Despite a near-continuum of possible monomial orders, most implementations rely on…
It often happens in mathematics that one and the same equation is known under different names in different areas of mathematics. The famous pentagon identity appears in low-dimensional topology in different ways. In this paper, we use the…
In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size $k \times k$ with non-negative…