Reducibility forces a rank drop of the closed-form coefficient matrix for normalized monic bivariate equations whose degrees up to d are nonzero in the base field, in particular below the characteristic and in characteristic zero. The rank drop is first derived from coprime factor decompositions, with coprimality imposed in the univariate fraction-field ring, where factorization can supply it.

theorem CCDN.closedForm_rank_drop_of_not_irreducible {F : Type u_1} [Field F] (d : ℕ) (hd : 0 < d) (hchar : CharZeroOrGT F d) (f : Polynomial (Polynomial F)) (hf : f.Monic) (hred : ¬Irreducible f) (hdeg : f.natDegree = d) (htotal : bivariateTotalDegree f ≤ d) : (closedFormCoefficientMatrix d f).rank + 2 ≤ 2 * d ^ 2

A reducible normalized equation of total degree at most d, with outer degree d, in characteristic zero or larger than d, has coefficient-matrix kernel dimension at least two. Both squarefree and repeated-factor cases are covered.