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In the proof of Theorem 3.7 of the original paper Internat. J. Math.30(14) (2019) 1950078, 1–25, two inequalities are used that do not hold in general. In this note, we prove an extra propositions which allows us to give a proof of Theorem 3.7 without using the aforementioned inequalities. Hence, all the results in the original paper are valid. We have also posted a corrected version in the arXiv. [ABSTRACT FROM AUTHOR]

In [J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61 (Princeton University Press, Princeton, NJ, 1968).] Milnor proved that a real analytic map f : (ℝ n , 0) → (ℝ p , 0) , where n ≥ p , with an isolated critical point at the origin has a fibration on the tube f | : 𝔹 𝜀 n ∩ f − 1 (𝕊 δ p − 1) → 𝕊 δ p − 1 . Constructing a vector field such that (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he "inflates" the tube to the sphere, to get a fibration φ : 𝕊 𝜀 n − 1 ∖ f − 1 (0) → 𝕊 p − 1 , but the projection is not necessarily given by f / ∥ f ∥ as in the complex case. In the case f has isolated critical value, in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and d -regularity for real analytic singularities, Internat. J. Math.21(4) (2010) 419–434.] it was proved that if the fibers inside a small tube are transverse to the sphere 𝕊 𝜀 , then it has a fibration on the tube. Also in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and d -regularity for real analytic singularities, Internat. J. Math.21(4) (2010) 419–434.], the concept of d -regularity was defined, it turns out that f is d -regular if and only if the map f / ∥ f ∥ : 𝕊 𝜀 n − 1 ∖ f − 1 (0) → 𝕊 p − 1 is a fiber bundle equivalent to the one on the tube. In a more general setting, the corresponding facts are proved in [J. L. Cisneros-Molina, A. Menegon, J. Seade and J. Snoussi, Fibration theorems and d -regularity for differentiable maps-germs with non-isolated critical value, Preprint (2017).], showing that if a locally surjective map f has a linear discriminant Δ with isolated singularity and a fibration on the tube f | : 𝔹 𝜀 n ∩ f − 1 (𝕊 δ p − 1 ∖ Δ) → 𝕊 δ p − 1 ∖ Δ , then f is d -regular if and only if the map f / ∥ f ∥ : 𝕊 𝜀 n − 1 ∖ f − 1 (Δ) → 𝕊 p − 1 ∖ 𝒜 (with 𝒜 the radial projection of Δ on 𝕊 p − 1 ) is a fiber bundle equivalent to the one on the tube. In this paper, we generalize this result for an arbitrary linear discriminant by constructing a vector field w ̃ which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that f / ∥ f ∥ is constant on the integral curves of w ̃. [ABSTRACT FROM AUTHOR]