The strength of a homogeneous polynomial (or form) is the smallest length an additive decomposition expressing it whose summands are reducible forms. Using functors, we show that set forms with bounded not always Zariski-closed. More specifically, if ground field algebraically closed, prove quartics ≤3 Zariski-closed for large number variables.

We factor the virtual Poincaré polynomial of every homogeneous space G/H , where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t(t − 1)QG/H(t ) for a polynomial QG/H with non-negative integer coefficients. Moreover, we show that QG/H(t ) divides the virtual Poincaré polynomial of every regular embedding of G/H , if H is connected. Introduction and statement ...

Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also morphisms πX : X → Z, πY : Y → Z. Given this data, we say that an object P of C, together with morphisms p1 : P → X, p2 : P → Y is a fiber product of X with Y over Z if it satisfies the following universal property: For every object T ∈ Obj(C), and every pair of morphisms f : T → X, g : T → Y such that πX ◦ f = πY ◦ g, ther...

We present a Zariski pair in affine complex plane consisting of two line arrangements, each of which has six lines. In the seminal paper [3], Zariski started the study of the fundamental groups of the complements of plane algebraic curves. Among other things, he constructed a pair of plane curves with the same degree and the same local singularities, but non-isomorphic fundamental groups, which...