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.

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...