The Properties of Topological Manifolds of Simplicial Polynomials

Abstract

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition.