  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 $P/I \longrightarrow K[X \setminus Z] / (I
\cap K[X \setminus Z])$ with maximal $\#Z$, are found degree-by-degree. They
turn out to yield optimal re-embeddings in the positively graded case. Viewing
$P_0 \longrightarrow P/I$ 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.
