Signed shape tilings of squares

Let T be a tile made up of finitely many rectangles whose corners have rational coordinates and whose sides are parallel to the coordinate axes. This paper gives necessary and sufficient conditions for a square to be tilable by finitely many Qweighted tiles with the same shape as T , and necessary and sufficient conditions for a square to be tilable by finitely many Z-weighted tiles with the same shape as T . The main tool we use is a variant of F.W.Barnes’s algebraic theory of brick packing, which converts tiling problems into problems in commutative algebra.