Parallel identity testing for skew circuits with big powers and applications

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labeled with powers [Formula: see text] for binary encoded numbers [Formula: see text]. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to [Formula: see text], which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in [Formula: see text]; this result is even new in the one-dimensional case, where the straight-line programs produce words. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products of finitely generated abelian groups belongs to [Formula: see text]. Using the Magnus embedding, it follows that the compressed word problem for a free metabelian group belongs to [Formula: see text].

A remark on spherical equations in free metabelian groups

I. Lysenok and A. Ushakov proved that the Diophantine problem for spherical quadric equations in free metabelian groups is solvable. The present paper proves this result by using the Magnus embedding.

THE MAGNUS EMBEDDING IS A QUASI-ISOMETRY

We show that the Magnus embedding, which embeds the free solvable group Sd, r of rank r and degree d into the wreath product ℤr ≀ Sd-1, r, is a quasi-isometry.

On the Magnus–Smelkin embedding

The generalization of the Magnus embedding [7] proved by Smelkin [9] may bestated as follows. Let L be a free group freely generated by the set xi(i∈I), and let R be a normal subgroup of L with G = L/R. If V is any variety of groups and ∏ is the V-freegroup with free generating set the symbols [g, xi] (g∈G, i∈I), then L/V(R) is embeddedin the semidirect product ∏ ⋊ G (where the action of G on ∏ is given by h · [g, xi] = [hg, xi], for h, g ∈ G).