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{\bf J. Bell, A.M. Garsia and N. Wallach}
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{\bf Some New Methods in the Theory of $m$-Quasi-Invariants}
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We introduce here a new approach to the study of
$m$-quasi-invariants. This approach consists in representing
$m$-quasi-invariants as $N^{tuples}$ of invariants. Then conditions
are sought which characterize such $N^{tuples}$. We study here the
case of $S_3$ $m$-quasi-invariants. This leads to an interesting free
module of triplets of polynomials in the elementary symmetric
functions $e_1,e_2,e_3$ which explains certain observed properties of
$S_3$ $m$-quasi-invariants. We also use basic results on finitely
generated graded algebras to derive some general facts about regular
sequences of $S_n$ $m$-quasi-invariants
\bye