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Syzygies for the vector invariants of the dihedral group

The problem of finding generators of the $GL$-ideal of the relations between the generators of the algebra of invariants of the dihedral group acting on $m$-tuples of vectors from its defining $2$-dimensional representation is studied. It is shown that this $GL$-ideal is generated by relations depending on no more than $3$ vector variables. A minimal $GL$-ideal generating system is found for the case when $m=2$, and for the case of the dihedral group of order $8$ and arbitrary $m$.