Diophantine equations in separated variables and polynomial power sums

We consider Diophantine equations of the shape $$ f(x) = g(y) $$ f ( x ) = g ( y ) , where the polynomials f and g are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many rational solutions (x, y) with a bounded denominator are only possible in trivial cases.

Diophantine equations in separated variables and lacunary polynomials

We study Diophantine equations of type [Formula: see text], where [Formula: see text] and [Formula: see text] are lacunary polynomials. According to a well-known finiteness criterion, for a number field [Formula: see text] and nonconstant [Formula: see text], the equation [Formula: see text] has infinitely many solutions in [Formula: see text]-integers [Formula: see text] only if [Formula: see text] and [Formula: see text] are representable as a functional composition of lower degree polynomials in a certain prescribed way. The behavior of lacunary polynomials with respect to functional composition is a topic of independent interest, and has been studied by several authors. In this paper, we utilize known results on the latter topic, and develop new ones, in relation to Diophantine applications.

Drinfeld center of planar algebra

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra P (not necessarily having finite depth). We prove that if N ⊂ M is a subfactor realization of P, then the Drinfeld center of the N–N-bimodule category generated byNL2(M)M, is equivalent to the category of Hilbert affine representations of P satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.

Filtration-preserving mappings and centralizers within graded Lie algebras

We use spectral sequence techniques to compute centralizers of elements within graded Lie algebras, and the methods are then applied to the calculation of unique normal forms of elements within one-parameter matrix Lie algebras. A finiteness criterion for unique normal forms is presented.