Z₂-equivariant standard bases for submodules associated with Z₂-equivariant singularities

Let x = (x₁,...,x_n) ∈ R ⁿ and λ ∈ R. A smooth map f(x,λ) is called Z₂-equivariant (Z₂-invariant) if f(−x, λ) = −f(x,λ) (f(−x, λ) = f(x,λ)). Consider the local solutions of a Z₂-equivariant map f(x,λ) = 0 around a solution, say f(x₀,λ₀), as the parameters smoothly vary. The solution set may experience a surprising behavior like observing changes in the number of solutions. Each of such problems/changes is called a singularity/bifurcation. Since our analysis is about local solutions, we call any two smooth map f(x,λ) and g(x,λ) as germ-equivalent when they are identical on a neighborhood of (x₀,λ₀) = (0,0). Each germ equivalent class is referred to a smooth germ. The space of all smooth Z₂-equivariant germs is denoted by [EQUATION] and space of all smooth Z₂-invariant germs is denoted by [EQUATION]_x,λ(Z₂). The space [EQUATION] is a module over the ring of Z₂-invariant germs [EQUATION]_x,λ(Z₂); see [3, 2, 7] for more information and the origins of our notations.