Divergence Zero Quaternionic Vector Fields And Hamming Graphs

We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs H(d, 2). We prove that monomials in variables z and w are always linearly independent as functions only in bidegrees (p, 0), (p, 1 ), (0, q), (1, q) and (2, 2).