On the rational relationships among pseudo-roots of a non-commutative polynomial

Abstract

For a non-commutative ring R , we consider factorizations of polynomials in R [ t ] where t is a central variable. A pseudo-root of a polynomial p ( t ) = p 0 + p 1 t + ⋯ p k t k is an element ξ ∈ R , for which there exist polynomials q 1 , q 2 such that p = q 1 ( t − ξ ) q 2 . We investigate the rational relationships that hold among the pseudo-roots of p ( t ) by using the diamond operations for cover graphs of modular lattices . When R is a division ring, each finite subset S of R corresponds to a unique minimal monic polynomial f S that vanishes on S . By results of Leroy and Lam [16] , the set of polynomials { f T : T ⊆ S } with the right-divisibility order forms a lattice with join operation corresponding to (left) least common multiple and meet operation corresponding to (right) greatest common divisor. The set of edges of the cover graph of this lattice correspond naturally to a set Λ S of pseudo-roots of f S . Given an arbitrary subset of Λ S , our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of Λ S , and in particular, rationally generates S .