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

Let SS 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 SS between two points s,t∈R2s,t∈R2, 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 (ACMQQ). In the ACMQQ, one can compute exactly any number that can be obtained from the rationals QQ by applying a finite number of operations from +, −, ×, ÷, ⋅k, for any integer k⩾2k⩾2. Our proof uses Galois theory and is based on Bajaj's technique.