Factorization in K[Xn, Xn+1,…,X2n−1]
David F. Anderson, Susanne Jenkens · Communications in Algebra · 1995
Let K be a field, n ≥ 2 an integer, and Rn = K[Xn, Xn+1,…,X2n−1]. In this paper, we study lengths of factorizations in Rn. For any atomic domain D, define ρ(D) = sup{ r/s ∣ X1…xr = y1…ys, xi, yj ∊ D irreducible } and φ(r) = ∣{ m ∣ X1 …xr = y1 …ym, xi, yj ∊ D irreducible }∣. We show that ρ(Rn)| =(nD(Gn(K)) + 3n − 2)/(2n), where D(Gn(K)) is the Davenport constant of an abelian group Gn(K) associated with K. Hence ρ(Rn) is finite if and only if K is finite. If K is finite, we also show that for Rn, .