The Weighted HOM-Problem over Fields

The HOM-problem, which asks whether the image of a regular tree language under a tree homomorphism is again regular, is known to be decidable. Since then, weighted versions of this problem for different semirings have also been investigated. In this paper, we prove the weighted HOM-problem for all fields decidable, provided that the tree homomorphism is tetris-free (a condition that generalizes injectivity). To this end, we reduce the problem to a property of the device representing the homomorphic image in question; to prove this property decidable, we then derive a pumping lemma for such devices from the well-known pumping lemma for regular tree series over fields, proved by Berstel and Reutenauer in 1982.