Related papers: Zero-dimensional tropicalizations in OSCAR
We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients…
We consider multidimensional optimization problems in the framework of tropical mathematics. The problems are formulated to minimize a nonlinear objective function that is defined on vectors over an idempotent semifield and calculated by…
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call `tropical polynomials' and `sign polynomials', respectively. We classify all irreducible polynomials in either case. We…
A motivation to study Gr\"{o}bner theory for fields with valuations comes from tropical geometry, for example, they can be used to compute tropicalization of varieties \citep{maclagan2009introduction}. The computational aspect of this…
We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…
We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We…
We suggest a version of Nullstellensatz over the tropical semiring, the real numbers equipped with operations of maximum and addition.
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…
Tropical Newton-Puiseux polynomials defined as piece-wise linear functions with rational coefficients at the variables, play a role of tropical algebraic functions. We provide explicit formulas for tropical Newton-Puiseux polynomials being…
We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free…
We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.
The usual approach to tropical geometry is via degeneration of amoebas of algebraic subvarieties of an algebraic torus $(\mathbb{C}^*)^n$. An amoeba is logarithmic projection of the variety forgetting the angular part of coordinates, called…
A tropical (or min-plus) semiring is a set $\mathbb{Z}$ (or $\mathbb{Z \cup \{\infty\}}$) endowed with two operations: $\oplus$, which is just usual minimum, and $\odot$, which is usual addition. In tropical algebra the vector $x$ is a…
We develop a probabilistic algorithm of Kronecker type for computing a Kronecker representation of a zero-dimensional linear section of an algebraic variety $V$ defined over a perfect field $k$. The variety $V$ is the Zariski closure of the…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
Decomposing an algebraic variety into irreducible or equidimensional components is a fundamental task in classical algebraic geometry and has various applications in modern geometry engineering. Several researchers studied the problem and…
Improved algorithms for computing (partial and full) exterior algebraic shifts of hypergraphs and simplicial complexes are presented. The main benefit is in positive characteristic. Experiments with an implementation in OSCAR with various…
A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph…