A generalized join theorem for real analytic singularities

Let f₁ : (Rⁿ, 0ₙ) → (R², 0₂) and f₂ : (Rᵐ, 0ₘ) → (R², 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 (Rⁿ⁺ᵐ, 0ₙ₊ₘ) to (R⁴, 0₄). We assume that f₁ and f₂ satisfy the af-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 (C², 0₄) to (C, 0₂). If we identify C with R², then g also defines a real analytic map germ from (R⁴, 0₄) to (R², 0₂). Then the real analytic map germ f : (Rⁿ × Rᵐ, 0ₙ₊ₘ) → (R², 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 0ₙ₊ₘ. 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.