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Let X be an affine toric variety. The total coordinates on X provide a canonical presentation
of X as a quotient of a vector space
by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.
We classify all finitely generated integral algebras with a rational action of a
reductive group such that any invariant subalgebra is finitely generated. Some results on
affine embeddings of homogeneous spaces are also given.
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