Quantum polynomial-time conjugacy in metabelian groups

We prove that the conjugacy problem in finitely presented metabelian groups with a fixed number of generators is solvable in bounded-error quantum polynomial time, uniformly in the defining relations of the group. Both the group presentation and the words representing the elements are part of the input. The main ingredient is a quantum polynomial-time algorithm for computing all multiplicative relations among a fixed number of units in a commutative affine $\mathbb Z$-algebra presented with a fixed number of variables. The algorithm reduces the problem to computations in finite-dimensional algebras over $\mathbb Q$ and finite rings by identifying the lattice of exponent vectors whose corresponding products are algebraic over the coefficient field. An effective arithmetic splitting combines the rational and finite-ring computations. The proof uses effective bounds for polynomial equations, algorithms for finite-dimensional commutative algebras, quantum integer factorization, and finite abelian hidden-subgroup algorithms.