International Journal of Mathematics

In this paper, all irreducible factors of bivariate polynomials of the form f(x) - g(y) over an arbitrary field are described. It also proves that the number of irreducible factors of f(x) - g(y) (counting multiplicities) does not exceed the greatest common divisor of the degrees of f(x) and g(y), yielding a well known result of Tverberg regarding the irreducibility of f(x) - g(y). It proves that if f(x) and g(y) are non-constant polynomials with coefficients in the field ℚ of rational numbers and degf(x) is a prime number, then f(x) - g(y) is a product of at most two irreducible polynomials over ℚ. This contributes to a problem raised by Cassels which asks for the polynomials f, such that the polynomial is reducible.

We study Milnor fibrations of real analytic maps , n ≥ p, with an isolated critical value. We do so by looking at a pencil associated canonically to every such map, with axis V = f^-1(0). The elements of this pencil are all analytic varieties with singular set contained in V. We introduce the concept of d-regularity, which means that away from the axis each element of the pencil is transverse to all sufficiently small spheres. We show that if V has dimension 0, or if f has the Thom a_f-property, then f is d-regular if and only if it has a Milnor fibration on every sufficiently small sphere, with projection map f/‖f‖. Our results include the case when f has an isolated critical point. Furthermore, we show that if f is d-regular, then its Milnor fibration on the sphere is equivalent to its fibration on a Milnor tube. To prove these fibration theorems we introduce the spherefication map, which is rather useful to study Milnor fibrations. It is defined away from V; one of its main properties is that it is a submersion if and only if f is d-regular. Here restricted to each sphere in ℝⁿ the spherefication gives a fiber bundle equivalent to the Milnor fibration.

We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres.

We prove that the images of irreducible germs of plane curves by a germ of analytic morphism φ have a certain contact either with branches of the discriminant of φ or with certain infinitesimal structures (shadows) that arise from the branches of the Jacobian of φ that are mapped to a point (and therefore give rise to no branch of the discriminant).

Let f be a positive smooth function on a closed Riemann surface (M, g). The f-energy of a map u from M to a Riemannian manifold (N, h) is defined as

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L^⊗r is trivial. This group Γ acts on by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

Let V be a vector space with Minkowski norm F. We find a geometric quantity on (V, F) which characterizes the Randers norm. We also give a different proof of the Matsumoto–Hōjō' theorem.

For finite dimensional abelian subalgebras of a finite von Neumann algebra, we obtain the value of conditional relative entropy defined by Choda. We also consider the modified invariant defined by Pimsner and Popa.