Irreducibility of random series and their small-order perturbations

Prove that for any field K of characteristic 0 and any generalized power series b in the ring K((ℝ^{≤0})) that is random—meaning either (i) the set cl(supp(b)) \ {0} is ℚ-linearly independent in ℝ or (ii) the coefficient tuple ⟨b_γ : γ ∈ supp(b)⟩ is algebraically independent over ℚ—and satisfies sup(supp(b)) = 0, the series b is irreducible in K((ℝ^{≤0})). Moreover, determine that for every series r ∈ K((ℝ^{≤0})) with ot(r) < ω^{deg(b)} (where ot(·) is the ordinal order type of the support and deg(b) is the Cantor degree of ot(b)), the series b + r is also irreducible.

Background

The paper studies irreducibility in rings of generalized power series K((ℝ^{≤0})) and develops large classes of explicit irreducible elements with prescribed ordinal order types. A central notion is that of a random series, defined via either algebraic independence of coefficients or ℚ-linear independence of (the closure of) exponents.

The authors prove broad irreducibility results for random principal series across many ordinal patterns, and extend these to certain non-principal series. They then formulate a conjecture that aims to cover all random series with support supremum 0, asserting irreducibility for such b and stability of irreducibility under adding any lower-order-type perturbation r.