A note on the unsolvability of the weighted region shortest path problem

Abstract Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s , t ∈ R 2 , where the distances are measured according to the weighted Euclidean metric—the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers ( ACM Q ). In the ACM Q , one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, −, ×, ÷, ⋅ k , for any integer k ⩾ 2 . Our proof uses Galois theory and is based on Bajaj's technique.