Signsectors

Signsectors are a concept in real algebraic geometry and computational geometry that describe the connected regions of a real space where a finite family of real-valued functions attains a fixed sign pattern. They provide a way to partition space according to the signs of the functions involved.

Definition: Let F = {f1, f2, ..., fm} be real-valued functions defined on R^n. A signsector is a maximal connected subset of R^n on which each function fi has a prescribed sign (positive or negative) and, in the most common formulation, none of the fi vanishes. More generally, one may consider regions where each fi has a specified sign or is allowed to be zero, with the boundaries lying on the zero sets Z(fi). The signvector associated to a point x is the m-tuple (sgn(f1(x)), sgn(f2(x)), ..., sgn(fm(x))).

Relation to other concepts: Signsectors are the cells of the arrangement formed by the hypersurfaces fi = 0 when the functions are polynomials or algebraic. They correspond to the regions where the sign conditions of the functions are constant, and they define parts of the semi-algebraic set determined by the sign conditions.

Properties and computation: For a finite family of polynomials, the signsectors partition the domain into open, connected components whose boundaries lie on the zero sets of the polynomials. The number of signsectors is finite within bounded regions and depends on m, n, and the degrees of the functions. Computing signsectors typically uses cell decomposition methods such as cylindrical algebraic decomposition or explicit arrangements of hypersurfaces, which can be computationally intensive in higher dimensions or with many polynomials.

Applications: Signsectors are used in real algebraic geometry, optimization, robotics, motion planning, and qualitative analysis, where understanding where constraints hold with certain signs is essential.

See also: semi-algebraic sets, sign conditions, CAD, arrangement of hypersurfaces.