Related papers: Tropicalization and irreducibility of Generalized …
We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of…
A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original…
We give explicit formulas as well as a quadratic time algorithm to solve (so called) generalized Vandermonde's systems of p linear equations and n variables. It allows in particular to find all (so called Lagrange's) interpolation polynoms…
We introduce tropically unirational varieties, which are subvarieties of tori that admit dominant rational maps whose tropicalisation is surjective. The central (and unresolved) question is whether all unirational varieties are tropically…
For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our…
We call a finite-dimensional K-algebra A geometrically irreducible if for all d, all connected components of the affine scheme of d-dimensional A-modules are irreducible. We show that the geometrically irreducible algebras without loops…
We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean $d$-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity.…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results…
We introduce the tropicalization of closed subschemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the results of Einslieder, Kapranov, Lind,…
Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to…
Each Gr\"obner stratum of a tropical variety is a connected set of points, all of which induce the same initial subscheme. The Gr\"obner stratification is a coarsening of the decomposition into Gr\"obner polyhedra, and has the advantage…
Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…
We develop an approach to study the irreducibility of generic complete intersections in the algebraic torus defined by equations with fixed monomials and fixed linear relations on coefficients. Using our approach we generalize the…
Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational…
Very simple sufficient conditions for the irreducibility of $f(X^n)$ over an arbitrary unique factorization domain $Z$ are established via a generalization of a well known theorem of A. Capelli.
In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
We prove that for almost square tensor product grids and certain sets of bivariate polynomials the Vandermonde determinant can be factored into a product of univariate Vandermonde determinants. This result generalizes the conjecture [Lemma…
We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.