A constructive algorithm for the solution of a 2-D coprime condition

Solutions to certain 2-D polynomial coprime conditions are obtained in this work. Given the 2-D polynomials f(x,y) and g(x,y), the problems are: (1) If f and g have no common zeros, to find u(x,y) and v(x,y) such that uf + vg = 1. (2) If f and g have no common zeros in ????C2, to find u(x,y) and v(x,y) such that uf + vg has no zeros in ??. The proofs of the existence of u and v are constructive and algebraic. Problem (2) has applications to 2-D feedback system design.