An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems

The paper’s Algorithm 1 establishes necessary-and-sufficient conditions for a weakly reversible, deficiency-zero (WR0) realization and proves uniqueness when it exists. The candidate solution mirrors the same linear-algebraic backbone (W = Y Aκ, kernel–cone partition, affine-independence implies unique decomposition, and a positive-kernel/circulation argument for weak reversibility) and arrives at the same existence/uniqueness conclusions. Differences are mostly stylistic: the model phrases the weak-reversibility step via nonnegative circulations, while the paper invokes the Kirchhoff-matrix kernel supported on terminal SCCs; both are equivalent in this setting.