Related papers: Combinatorics of line arrangements and polynomial …

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…

Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…

Consider a finite dimensional vector space $V$ over a finite field $\mathbb{F}_q$. We give a minimal generating set for the ring of invariants $\mathbb{F}_q[V \oplus V^*]^{\text{GL}(V)}$, and show that this ring is a Gorenstein ring but is…

For a smooth algebraic variety $X$, we study the category of finitely generated modules over the ring of function of $X$ that has a compatible action of the Lie algebra $\mathcal{V}$ of polynomials vector fields on $X$. We show that the…

We investigate the rings of semi-invariants for tame string algebras A(n) of non-polynomial growth. We are interested in dimension vectors of band modules. We use geometric technique related to the description of coordinate rings on…

For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal…

In this paper we give full classification of rank 3 line arrangements in $\mathbb P^2$ (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to…

The boundary of the Milnor fiber associated with a complex line arrangement is a three dimensional plumbed manifold, and it is a combinatorial invariant. We prove the reverse implication, which was conjectured N\'emethi and Szil\'ard. That…

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincar\'e polynomial. We prove a Terao-type factorization statement on the splitting of…

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…

We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability…

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…

The enumeration of combinatorial classes of the complex polynomial vector fields in C presented in [Dia13] is extended here to a closed form enumeration of combinatorial classes for degree d polynomial vector fields up to rotations of…

Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincar\'e-Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in the complex projective plane. Such pair of arrangements has an additional property: they admit conjugated equations on the ring…

We consider an arbitrary representation of the additive group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.

We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the…

In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.