Multiplicative Dependence of Shifted Quadratic and Cubic Algebraic Numbers

We study short multiplicative relations among additive translates of algebraic numbers. Every quadratic algebraic number admits infinitely many relations with three factors on each side and arbitrarily large integer shifts. For quadratic algebraic integers, we classify the minimum total length over integer tails and show that it is eventually five. We also construct infinitely many reduced relations of length eight for every cubic algebraic number using rational shifts, and prove that the integer-shift length of 23 is exactly eight.