Elimination by substitution

Abstract

Let K be a field and $P=K[x_1,\dots ,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots ,z_s)$-separating tuple of polynomials $(f_1,\dots ,f_s)$ in an ideal I, i.e., on finding polynomials such that $f_i=z_i-h_i$ with $h_i\in K[X\setminus Z]$. 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 $P_0$-module membership problems. Best separable re-embeddings, i.e., isomorphisms $P/I⟶K[X\setminus Z]/(I\cap K[X\setminus Z\left]\right)$ with maximal #Z, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing $P_0⟶P/I$ 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.