Irreducibility of random series and stability under lower-order perturbations

Establish that for any field K of characteristic 0 and any random generalized power series b in K((R^{≤0})) with sup(supp(b)) = 0, the series b is irreducible in K((R^{≤0})), and moreover b + r is irreducible for every series r whose order type satisfies ot(r) < ω^{deg(b)}, where deg(b) denotes the Cantor degree of the order type of b and “random” means that either cl(supp(b))\{0} is Q-linearly independent in R or the tuple of coefficients ⟨b_γ : γ ∈ supp(b)⟩ is algebraically independent over Q.

Background

The paper studies irreducibility of generalized power series in the ring K((R^{≤0})) and develops broad classes of irreducible elements by leveraging principal series, ordinal valuations, and independence notions. A key notion is that of a “random” series, where either the exponents (excluding 0) form a Q-linearly independent set or the coefficients are algebraically independent over Q.

The authors prove large families of irreducible series (including non-principal ones) for many ordinals, and they introduce a hereditary rv_J-independence condition that guarantees irreducibility in their main theorems. Motivated by these results, they formulate a conjecture asserting that randomness alone should suffice to ensure irreducibility, and that irreducibility persists after adding any series whose order type is strictly below ω^{deg(b)}.