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

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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)=p0+p1t+⋯pktk is an element ξ∈R, for which there exist polynomials q1,q2 such that p=q1(t−ξ)q2. 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 fS that vanishes on S. By results of Leroy and Lam [16], the set of polynomials {fT: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 ΛS of pseudo-roots of fS. Given an arbitrary subset of ΛS, our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of ΛS, and in particular, rationally generates S.

Original languageEnglish (US)
Article number106581
JournalJournal of Pure and Applied Algebra
Issue number6
StatePublished - Jun 2021

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


  • Factorizations
  • Lattices
  • Non-commutative polynomials
  • Roots


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