Trinitary algebras

A { trinitary algebra} is any subalgebra of the space of smooth functions f: M → R that is distinguished in this space by finitely many linear conditions of the form f(xᵢ) = f( xᵢ) = f( xᵢ), where xᵢ, xᵢ, and xᵢ are distinct points in M, i=1, , k, or is approximated by such subalgebras. Trinitary algebras naturally arise in the study of discriminant varieties, i.e., the spaces of singular geometric objects, in theories where the property of being singular is formulated in terms of simultaneous behavior at three distinct points. The space of all trinitary algebras in C^∞(M, R) is analogous to the space of all ideals of finite codimension, which plays the same role in the study of discriminants defined in the terms of a single singular point, and also to the space of { equilevel algebras} (see {EA}) which are defined by collections of conditions f(xᵢ)=f( xᵢ) and play the same role in the study of discriminants defined by binary singularities like the self-intersections of the image. We classify the trinitary algebras up to the codimension four in C^∞(S¹, R), describe the cell structures of the varieties of these algebras, and calculate their homology groups and Stiefel--Whitney classes of their canonical normal bundles.