Elimination by Substitution

Let K be a field and P=K[x₁,,xₙ]. The technique of elimination by substitution is based on discovering a coherently Z=(z₁,,zₛ)-separating tuple of polynomials (f₁,,fₛ) in an ideal I, i.e., on finding polynomials such that fᵢ = zᵢ - hᵢ with hᵢ ∈ K[X Z]. Here we elaborate on this technique in the case when P is non-negatively graded. The existence of a coherently Z-separating tuple is reduced to solving several P₀-module membership problems. Best separable re-embeddings, i.e., isomorphisms P/I K[X Z] / (I ∩ K[X Z]) with maximal $#Z$, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing P₀ P/I as a fibration over an affine space, we show that its fibers allow optimal Z-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of P such that additional Z-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between $P/I$ and a non-negatively graded polynomial ring if $P/I$ is regular.