signsector

Signsector is a term used in some mathematical discussions to describe a region of a domain where a specified collection of real-valued functions attains a fixed sign pattern. It generalizes the idea of a one-dimensional sign chart to higher dimensions by partitioning input space according to the signs of several functions rather than a single function.

Formal definition: Let P = {p1, ..., pm} be real-valued functions defined on a domain D ⊆ R^n. For a sign vector σ ∈ {+,−}^m, the signsector S_σ is defined as S_σ = { x ∈ D | sign(p_i(x)) = σ_i for i = 1,...,m }. The treatment of zeros varies by source; in many contexts zeros are excluded, with the corresponding boundary described by some p_i(x) = 0.

Properties: If the functions p_i are polynomials, each S_σ is a semialgebraic set. The sectors form a (coarsely) disjoint partition of D up to boundary, and their boundaries lie in the union of the zero sets {x | p_i(x) = 0}. For linear functions, sectors are convex polyhedra; for general polynomials they can be non-convex and curved.

Examples: With p1(x,y) = x and p2(x,y) = y^2 − 1 on D = R^2, the signsectors correspond to regions where x is positive or negative and where y^2 is greater than or less than 1, separated by the lines x = 0 and the curve y = ±1.

Applications and relation: The concept supports sign-condition approaches in real algebraic geometry, constraint solving, and piecewise-defined modeling. The term is not universally standardized; related ideas include sign charts, sign conditions, and cell decompositions.