pointsforms
Pointsforms are an algebraic construct used to encode or encode-and-encode-like configurations of points within a geometric or algebraic framework. In its common form, a pointsform is a multivariate polynomial or polynomial-like object associated with a finite set of points S = {p1, ..., pk} in a space such as Euclidean space R^n. A pointsform can be designed to vanish on the points of S, to take prescribed values at those points, or to interpolate a given pattern of values across S. A canonical example is the product form F_S(x) = ∏_{i=1}^k ||x − p_i||^2 in R^n, which vanishes exactly at the points in S and is positive elsewhere. More general constructions use interpolation principles to produce forms that match specified data at S, analogous to Lagrange interpolation in several variables.
Properties commonly discussed include degree and the algebraic variety defined by the pointsform, which reflect the
Construction methods range from direct product formulas to solving Vandermonde-type systems, and from symmetric polynomials to
Applications span geometric hashing, pattern recognition, algebraic statistics, solving polynomial systems with known zero sets, and
See also: polynomial interpolation, vanishing ideals, Lagrange interpolation, algebraic geometry, geometric hashing.