In this work, we explore a non-linear extension of the classical Sylvester-Gallai configuration. Let \mathbb{K} be an algebraically closed field of characteristic zero, and let \mathcal{F} = \{F_1, \ldots, F_m\} \subset \mathbb{K}[x_1, \ldots, x_N] denote a collection of irreducible homogeneous polynomials of degree at most d, where each F_i is ... more >>>

We prove a sensitivity-to-communication lifting theorem for arbitrary gadgets. Given functions f: \{0,1\}^n\to \{0,1\} and g : \mathcal{X} \times \mathcal{Y}\to \{0,1\}, denote f\circ g(x,y) := f(g(x_1,y_1),\ldots,g(x_n,y_n)). We show that for any f with sensitivity s and any g,

where ... more >>>

A central question in mathematics and computer science is the question of determining whether a given ideal I is prime, which geometrically corresponds to the zero set of I, denoted Z(I), being irreducible. The case of principal ideals (i.e., m=1) corresponds to the more familiar absolute irreducibility testing of polynomials, ... more >>>