Unique factorization results for generalized power series

Abstract

Factorization in rings of the form $K((G^{\le 0} )) $ exhibits pathological behavior due to divisibility by monomials. It remains open whether this is the only obstruction and whether the quotient by the ideal generated by all monomials is a unique factorization domain. Prior to this work, the only instances of unique factorization beyond the irreducible elements followed directly from the primality of elements which aren't divisible by any monomial and whose support has order type $\omega$ or $\omega+1$. Using the $RV$ tool from valuation theory as a key ingredient, we establish unique factorization for elements of $ K((\mathbb{R}^{\le 0}))$ which aren't divisible by any monomial and whose support has order type $\omega^{\alpha} $ or $\omega^{\alpha}+1 $ for a large class of non additively principal ordinals $\alpha$.