A generalized join theorem for real analytic singularities

Let f₁: (R^n, 0₍) (R^2, 0₂) and f₂: (R^m, 0₌) (R^2, 0₂) be real analytic map germs of independent variables, where n, m 2. Then the pair (f₁, f₂) of f₁ and f₂ defines a real analytic map germ from (R^n+m, 0₍+₌) to (R^4, 0₄). We assume that f₁ and f₂ satisfy the a₅-condition at 0₂. Let g be a strongly non-degenerate mixed polynomial of 2 complex variables which is locally tame along vanishing coordinate subspaces. A mixed polynomial g defines a real analytic map germ from (C^2, 0₄) to (C, 0₂). If we identify C with R^2, then g also defines a real analytic map germ from (R^4, 0₄) to (R^2, 0₂). Then the real analytic map germ f: (R^n R^m, 0₍+₌) (R^2, 0₂) is defined by the composition of g and (f₁, f₂), i. e. , f (x, y) = (g (f₁, f₂) ) (x, y) = g (f₁ (x), f₂ (y) ), where (x, y) is a point in a neighborhood of 0₍+₌. In this paper, we first show the existence of the Milnor fibration of f. We next show a generalized join theorem for real analytic singularities. By this theorem, the homotopy type of the Milnor fiber of f is determined by those of f₁, f₂ and g. For complex singularities, this theorem was proved by A. Némethi. As an application, we show that the zeta function of the monodromy of f is also determined by those of f₁, f₂ and g.