EXPLICITLY SOLVABLE SYSTEMS OF FIRST-ORDER DIFFERENCE EQUATIONS WITH HOMOGENEOUS POLYNOMIAL RIGHT-HAND SIDES

The paper’s Proposition asserts that the discrete homogeneous system admits the closed form zn(s) = [zn(0)]^{M^s} Z^{(M^s − 1)/(M − 1)} under the N constraints Z = ∑_{|m|=M} cn,m ∏k (zk(0)/zn(0))^{m_k} and claims this is verified by direct substitution. However, substituting the proposed form yields RHS = zn(s)^M · Hn((r^{(n)})^{M^s}), with Hn the homogeneous ratio-polynomial. Equality with zn(s+1) requires Hn((r^{(n)})^{M^s}) ≡ Z for all s ≥ 0, whereas the constraints enforce only Hn(r^{(n)}) = Z at s = 0. The paper’s proof overlooks this propagation requirement, so the proposition fails in general. A concrete quadratic 2D counterexample satisfying the constraints at s = 0 matches the s = 1 step but already fails at s = 2, contradicting the claimed solution. The paper’s stated formula and constraints appear in the Proposition and example sections of the PDF (see Proposition and (4a)–(4b) and the N = 2, M = 4 example) .