Square matrices represent linear self-maps of vector spaces, and their eigenpoints are the fixed points of the induced map on projective space. Likewise, polynomial self-maps of a projective space are represented by tensors. We study the configuration of fixed points of a tensor or symmetric tensor.1991 Mathematics Subject Classification. Primary 15A18; Secondary 13P25, 14M12, 15A69.
Abstract. V. I. Arnold proved in 1991Arnold proved in (published in 1993 that the intersection multiplicity between two germs of analytic subvarieties at a fixed point of a holomorphic invertible self-map remains bounded when one of the germs is dragged by iterations of the self map. The proof is based on the SkolemMahler-Lech theorem on zeros in recurrent sequences.We give a different proof, based on the Noetherianity of certain algebras, which allows to generalize the Arnold's theorem for local actions of arbitrary finitely generated commutative groups, both with discrete and infinitesimal generators. Simple examples show that for non-commutative groups the analogous assertion fails.
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