REVIEW 3 major objections 5 minor 2 references
The Groebner basis and solution set of a polynomial system related to the Jacobian conjecture
T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read For a family of polynomial systems tied to the two-dimensional Jacobian conjecture, the paper reduces the system to two equations and describes every solution explicitly.
desk verdict The triangular Groebner basis result is new and likely correct, but the solution-set parametrization is wrong as printed, including a nonsensical count in the odd case. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is a homogeneous weighted term order: the matrix wmat induces a degree reverse lexicographic monomial order, a comparison that first looks at total weight and breaks ties by reversing lexicographic order, with weights $w(C_{-i})=i+1$ and $w(y)=m+2$. Under this order every generator $E_k$ has leading term $3C_{-(k+2)}$, and because these leading terms are pairwise coprime monomials, Buchberger's criterion verifies that $\{E_1,\ldots,E_{m-1}\}$ is a Groebner basis. The reduction algorithm then produces the triangular basis $\tilde E_k = C_{-(k+2)} + R_k(C_{-1},C_{-2})$. In the mixed cases the substitution $t = C_{-2}^2/C_{-1}^3$ reduces the final two equations to the univariate polynomials $f(t)$ and $g(t)$, which carry the root-counting and parametrization of the solution set.
What would settle it
Compute the reduced Groebner basis of $I_{m-1}$ with the stated weighted order for a small case such as $m=4$ or $m=5$ using a computer algebra system; if any S-polynomial fails to reduce to zero, or any reduced basis polynomial has a leading term other than $C_{-(k+2)}$, Proposition 2.1 is false. Alternatively, for $m=8$ with $C_{-2}=0$, solve the system directly: if a solution exists despite the paper's $m\equiv 1 \bmod 3$ condition in that case, the stated case classification is incomplete.
Extended reading notes
Core claim
Proposition 2.1 is the central claim: with the weighted degree reverse lexicographic order defined by the matrix wmat (weights $w(C_{-i})=i+1$ and $w(y)=m+2$), the set $\{E_1,\ldots,E_{m-1}\}$ is a Groebner basis of $I_{m-1}$, and its reduced basis has the form $\tilde E_k = C_{-(k+2)} + R_k(C_{-1}, C_{-2})$ with $R_k$ homogeneous of weight $k+3$. The coefficient identity $E_k = 3C_{-(k+2)} + T(C_{-1},\ldots,C_{-k})$ ensures that the leading terms are pairwise coprime monomials, so Buchberger's criterion applies. Dividing the last two generators by this basis shows that the full system is equivalent to $\tilde E_m = R_m(C_{-1},C_{-2})=0$ and $\tilde E_{m+1} = y + R_{m+1}(C_{-1},C_{-2})=0$. The paper then classifies all solutions: $C_{-1}=0$ forces $m\equiv 1 \bmod 3$ and yields $j=(m+2)/3$ solutions; $C_{-2}=0$ forces $m$ even and yields $i=(m+2)/2$ solutions; and if both are nonzero, the substitution $C_{-2}^2 = t C_{-1}^3$ turns the two equations into univariate polynomials $f(t)$ and $g(t)$ whose roots parametrize all solutions, with at most $s(m+2)$ solutions for even $m$ and at most $2s(m+2)$ for odd $m$.
Load-bearing premise
The whole reduction rests on the assumption that, under the chosen weighted ordering, each equation $E_k$ has leading monomial $C_{-(k+2)}$, and on the cited fact that the first $m+n-2$ coefficients determine all later coefficients; if either assumption fails, the triangular basis and the two-equation reduction collapse.
Editorial extensions
If this is right
- Every solution of the specialized system is determined by the two coefficients $C_{-1}, C_{-2}$; the remaining coefficients $C_{-3},\ldots,C_{-(m+1)}$ are computed as $-R_k(C_{-1},C_{-2})$.
- For each fixed $y$, the solution set is finite with explicit bounds: $j=(m+2)/3$ solutions in the first case, $i=(m+2)/2$ in the second, and at most $s(m+2)$ or $2s(m+2)$ in the mixed cases according to the parity of $m$.
- Because the ideal $I_{m-1}$ has a triangular Groebner basis, the variety it defines is a complete intersection with coordinates $C_{-1}, C_{-2}$, giving a concrete normal form for the quotient ring.
- Within the Jacobian-conjecture framework, any solution of these reduced equations supplies the data $(P,Q,C,F)$ of a potential counterexample, so the parametrization yields an explicit finite search space for such data.
Reading between the lines
- The same weighted-order strategy plausibly extends to $n>3$: the leading term of $E_k$ would be $C_{-(k+n-1)}$, and the reduced system would be governed by the first $n-1$ coefficients, shrinking the Jacobian-conjecture system to $n-1$ variables; this is an extrapolation, not a claim of the paper.
- The polynomials $R_k$, though not given in closed form, satisfy a recursive triangular division; computing them symbolically for larger $m$ would make the polynomials $f(t)$ and $g(t)$ explicit and allow direct verification of the solution counts.
- If the printed condition $m\equiv 1 \bmod 3$ in the $C_{-2}=0$ case is taken literally, then even values $m\equiv 2 \bmod 3$ would be excluded; a direct Groebner or substitution computation for $m=8$ would settle whether the classification is missing a family of solutions or the printed congruence is a typo.
- The solution parametrization implies that, for generic $y$, the number of solutions depends only on the number of distinct roots of $f$ and on the parity of $m$, a prediction that could be tested numerically for small $m$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies a polynomial system in variables C_{-1},...,C_{-(m+1)}, y that arises from the Guccione–Guccione–Valqui system of equations equivalent to the existence of a counterexample to the two-dimensional Jacobian conjecture. For n=3, 3∤m, ν_i=0 (i>0), and F_{1-n}=y, the authors compute a Gröbner basis for the ideal I_{m-1} generated by the first m-1 equations (Proposition 2.1). They obtain a triangular reduced Gröbner basis \tilde E_k = C_{-(k+2)} + R_k(C_{-1},C_{-2}), k=1,...,m-1. The remaining two equations reduce to \tilde E_m = R_m(C_{-1},C_{-2})=0 and \tilde E_{m+1}=y+R_{m+1}(C_{-1},C_{-2})=0. Section 3 then attempts to describe all solutions of this two-equation system by splitting into four cases according to whether C_{-1}, C_{-2} vanish and according to the parity of m, leading to Propositions 3.2, 3.4, 3.5, and 3.6.
Significance. The Gröbner basis part (Proposition 2.1) is a clean structural result: the triangular form reduces a system of m+1 equations in m+1 unknowns to two polynomial equations in two unknowns, with the remaining variables eliminated explicitly. This is a useful reduction, and the proof via Buchberger's criterion with pairwise coprime leading terms is standard. The subsequent univariate reductions in Section 3 are a natural and potentially fruitful way to count solutions. However, the solution-set analysis as printed contains serious errors in the odd-m case (Proposition 3.6) and an internal inconsistency in Proposition 3.4. Since the abstract claims a complete description of the solution set, these errors are load-bearing. The paper also leaves open the possibility that the auxiliary polynomial f(t) is identically zero, which would invalidate the claimed finite bounds. If the odd case is repaired and the degeneracies are addressed, the result would be a worthwhile contribution to the algebraic study of this Jacobian-conjecture system.
major comments (3)
- [§3, Proposition 3.6] The odd-m parametrization is not mathematically valid. After substituting C_{-2}^2 = t C_{-1}^3, equation (3.8) becomes y + C_{-1}^{(m+2)/2} t^{1/2} g(t)=0. Since m is odd, the exponent (m+2)/2 is half-integral, so the formula C_{-1} = (-y/(t^{1/2}g(t)))^{2/(m+2)} and the statement '(m+2)/2 choices' have no well-defined meaning. The correct treatment is to set u=C_{-1}^{1/2}; then u^{m+2} = -y/(t^{1/2}g(t)) has m+2 solutions, and each gives C_{-1}=u^2 and C_{-2}=t^{1/2}u^3. The identification (u,t^{1/2}) ∼ (-u,-t^{1/2}) shows there are m+2 distinct pairs (C_{-1},C_{-2}) per root t, not 2(m+2). For example, for m=5 equations (3.7)-(3.8) reduce to λ_{30}C_{-1}^3+λ_{02}C_{-2}^2=0 and y+λ_{21}C_{-1}^2C_{-2}=0, which imply C_{-1}^7 is constant and yield 7 solutions in the nonzero case, not 14. The bound 'at most 2·s·(m+2)' and the displayed parametrization are therefore wrong.
- [§3, Proposition 3.4] The first bullet states m≡1 mod 3, but the derivation in the second case requires m even (from 2i=m+2), and Lemma 3.3 explicitly assumes 2|m. For an odd m with m≡1 mod 3, e.g. m=7, the quantity i=(m+2)/2 is not an integer, so the statement 'There are i solutions' is nonsensical. The congruence should read m even (or, using 3∤m, m≡4 mod 6). This typo makes the proposition internally inconsistent and would mislead a reader applying the case distinction.
- [§3, Propositions 3.5 and 3.6] The 'at most' bounds assume that the polynomial f(t) defined in (3.13) and (3.14) has exactly s roots, so f is nonzero. The paper never proves that R_m(C_{-1},C_{-2}) is not the zero polynomial. If R_m ≡ 0, then f ≡ 0 and the variable t becomes free, potentially giving infinitely many solutions via (3.8); the claimed finite description would then fail. The degenerate cases g(t_l)=0 or y=0 are also not discussed. Since the paper claims a complete description, these possibilities should be ruled out or explicitly excluded.
minor comments (5)
- [Title] The spelling 'Groebner' should be 'Gröbner'.
- [After (2.1)] The phrase '(C^2)_{-i}' should read '(C^3)_{-i}'.
- [Proof of Proposition 2.1] The assertion that LT(E_k)=C_{-(k+2)} in the given monomial order is not justified; a short explanation that all monomials of E_k have weight k+3 and that the grevlex tie-breaking makes C_{-(k+2)} the leader would improve the exposition.
- [Propositions 3.5 and 3.6] The expression 'm+2/2 choices' is ambiguous; it should be written as (m+2)/2.
- [Section 3] The paper does not provide the polynomials R_m and R_{m+1} for any m, nor does it indicate how they can be computed; since the solution formulas depend on the constants λ, a few explicit small-m examples or an algorithm would aid verification.
Circularity Check
No circular derivation: the Groebner-basis and solution-set computation is self-contained given the inherited system; self-citations are background, not load-bearing.
full rationale
The paper's derivation chain is self-contained. The system (2.1) is inherited from [2, Theorem 1.9], but the actual results—Proposition 2.1's triangular reduced Groebner basis and the solution-set enumeration via (3.5)-(3.8)—are derived from the polynomial definitions and a specified weighted monomial order, not from any assumption of the Jacobian conjecture. No parameter is fitted to data and then renamed a prediction; no quantity is defined in terms of the claimed output; no ansatz is smuggled in via citation. The citation to [2, Remark 1.13] for coefficient-determinacy is a parameter-free triangular lemma that the paper also states as 'easy to see'; even if it were considered load-bearing, it is prior mathematical content, not an assumption of the result being derived. The cited [1] is external textbook material. The manuscript does contain verification gaps that are correctness, not circularity, concerns: the proof of Proposition 2.1 asserts the leading-term identity E_k = 3C_{-(k+2)} + ... without a detailed check under wmat, and Proposition 3.6's odd-case count (m+2)/2 is non-integer as written and appears inconsistent with a direct u^{m+2} count. Neither gap makes the output equivalent to the input by construction. Score 0.
Assumptions & free parameters
assumptions (3)
- domain assumption The first m+n-2 coefficients C_{-1},...,C_{-(m+n-2)} determine all higher coefficients C_{-k} of a solution C.
- domain assumption Theorem 1.1 equivalence between the Jacobian conjecture being false and existence of C, P, Q, F with the stated properties.
- standard math Buchberger criterion and Proposition 2.9.4 of Cox-Little-O'Shea: S-polynomials of polynomials with relatively prime leading terms reduce to zero.
Cite this review
Pith. "Pith review of The Groebner basis and solution set of a polynomial system related to the Jacobian conjecture." pith.science (2026). https://pith.science/paper/IBCY3PAG
@misc{pith2026250605697,
author = {Pith},
title = {Pith review of: The Groebner basis and solution set of a polynomial system related to the Jacobian conjecture},
year = {2026},
howpublished = {\url{https://pith.science/paper/IBCY3PAG}},
note = {Machine review of arXiv:2506.05697}
}
read the original abstract
We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture, and describe completely the solution set.
Reference graph
Works this paper leans on
-
[1]
An introduction to computational algebraic geometry and commu- tative algebra
David Cox, John Little, and Donal O’Shea,Ideals, varieties, and algorithms, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. An introduction to computational algebraic geometry and commu- tative algebra
work page 2007
-
[2]
A system of polynomial equations related to the Jacobian Conjecture
Jorge Alberto Guccione, Juan Jos´ e Guccione, and Christian Valqui,A system of polynomial equations related to the Jacobian conjecture, arXiv:1406.0886v1 [math.AG] (3 June 2014). Christian V alqui, Pontificia Universidad Cat´olica del Per´u, Secci´on Matem´aticas, PUCP, A v. Uni- versitaria 1801, San Miguel, Lima 32, Per ´u. Instituto de Matem ´atica y Ci...
work page Pith review arXiv 2014
Reviewed August 7, 2026 · model on record in the stance chip above.
Discussion (0). Sign in to comment.