Hypergraphs and Lotka-Volterra systems with linear Darboux polynomials

Peter H. van der Kamp

correctlow confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The candidate reproduces the paper’s tree-based parametrisation of the solutions to (C3), including the nondegeneracy assumption (Ax,y−Ay,y)(Ax,x−Ay,x)≠0 on edges, and derives exactly the same formulas αj/αh as a product of edge ratios along T and the expression for non-tree entries Ax,y via the unique path product (the paper’s (10)–(11)). The proof structure—splitting c(e)=0 into tree edges to determine α up to scale and non-tree edges to impose linear relations on A, then propagating ratios along paths—is identical to Proposition 3’s argument, and the parameter/dimension count matches Proposition 4 (plus one overall scale for α). No missing hypotheses beyond αj≠0 and the edge nondegeneracy are required. See the statements and proofs around (C3), (9)–(12) in Proposition 3/4 in the PDF .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper’s core result—parametrising the general solutions to (C3) via a spanning tree with simple path-product formulas—is correct and presented clearly. It is immediately useful for constructing classes of LV-systems with linear Darboux polynomials and underpins the later equivalence/classification work. Minor clarifications would further improve readability, particularly around parameter counts and the explicit role of the overall scaling in α, but no substantive changes are needed.