Rational elasticity of factorizations in Krull domains

Abstract

For an atomic domain R R , we define the elasticity of R R as ρ ( R ) = sup ( m / n | x 1 ⋯ x m = y 1 ⋯ y n , for x i , y j ∈ R irreducibles}  \rho (R) = \sup (m/n|{x_1} \cdots {x_m} = {y_1} \cdots {y_n},\;{\text {for}}\;{x_i},{y_j} \in R\;{\text {irreducibles\} }} and let l R ( x ) {l_R}(x) and L R ( x ) {L_R}(x) denote, respectively, the inf and sup of the lengths of factorizations of a nonzero nonunit x ∈ R x \in R into the product of irreducible elements. We answer affirmatively two rationality conjectures about factorizations. First, we show that ρ ( R ) \rho (R) is rational when R R is a Krull domain with finite divisor class group. Secondly, we show that when R R is a Krull domain, the two limits l R ( x n ) / n {l_R}({x^n})/n and L R ( x n ) / n {L_R}({x^n})/n , as n n goes to infinity, are positive rational numbers. These answer, respectively, conjectures of D. D. Anderson and D. F. Anderson, and D. F. Anderson and P. Pruis. (The second question has also been solved by A. Geroldinger and F. Halter-Koch.)