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 is an element of K[X\Z]. Here we elaborate on this technique in the case when Pis nonnegatively 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 boolean AND 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. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
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