Difference between revisions
moh-degree-100
Status and infobox
- status:
—→ proved - tier:
—→ V3 - kind:
—→ theorem - year:
—→ 1983 - statement:
—→ If $f,g\in\mathbb C[x,y]$ have constant nonzero Jacobian and $\max(\deg f,\deg g)\le100$, then $\mathbb C[f,g]=\mathbb C[x,y]$. - short:
—→ JC₂ holds when max(deg f, deg g) ≤ 100 (computer-assisted)
Relations
- + special case of jc-plane: He then ran a computer search over degree pairs up to 100, which left six possible cases, and eliminated those separately. The result is that the plane conjecture holds for every Jacobian pair of degree at most 100.
Source +22 −0
+---
+title: Moh's theorem: no plane counterexample of degree ≤ 100
+short: JC₂ holds when max(deg f, deg g) ≤ 100 (computer-assisted)
+kind: theorem
+status: proved
+tier: V3
+year: 1983
+authors: T. T. Moh
+statement: If $f,g\in\mathbb C[x,y]$ have constant nonzero Jacobian and $\max(\deg f,\deg g)\le100$, then $\mathbb C[f,g]=\mathbb C[x,y]$.
+---
+Moh analysed how the Puiseux roots at infinity of a hypothetical Jacobian pair can be configured. He then ran a computer search over degree pairs up to 100, which left six possible cases, and eliminated those separately.<ref>T. T. Moh, On the Jacobian conjecture and the configurations of roots, J. reine angew. Math. 340 (1983) 140–212, [doi:10.1515/crll.1983.340.140](https://doi.org/10.1515/crll.1983.340.140).</ref> The result is that [[special-case-of::Statement:jc-plane|the plane conjecture holds for every Jacobian pair of degree at most 100]].
+The proof has a documented history, and the atlas records all of it.
+
+- **2005.** L.-C. Wang recoded the algorithm, found exceptions beyond Moh's six cases, and modified it with approximate-root theory “to fulfill his claim”.<ref>L.-C. Wang, On the Jacobian conjecture, Taiwanese J. Math. 9 (2005) 421–431, [doi:10.11650/twjm/1500407850](https://doi.org/10.11650/twjm/1500407850).</ref>
+- **2016–2022.** Xu finds an unhandled split of minor roots and suggests that the case (99, 66) is open (unrefereed).<ref>Y. Xu, [Intersection Numbers and Split of Minor Roots](https://arxiv.org/abs/1604.07683), arXiv:1604.07683 (v4 2022).</ref>
+- **2022.** Guccione, Guccione, Horruitiner and Valqui, with their own case analysis, discard every pair with maximum degree below 125 except (72, 108). They thereby confirm Moh's bound and raise it to 108 (unrefereed).<ref>J. A. Guccione, J. J. Guccione, R. Horruitiner, C. Valqui, [Increasing the degree of a possible counterexample to the Jacobian Conjecture from 100 to 108](https://arxiv.org/abs/2204.14178), arXiv:2204.14178.</ref>
+- **2025.** T. Nguyen proves the complex case up to degree 104. The proof takes Moh's bound as input, together with the known gcd results.<ref>T. Nguyen, Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104, Quaest. Math. 48 (2025) 1291–1305, [doi:10.2989/16073606.2025.2482655](https://doi.org/10.2989/16073606.2025.2482655).</ref>
+
+So the refereed bound is 104, the claimed bound is 108, and one case of the original argument is disputed in an unrefereed preprint. The disputed case lies inside the range that the unrefereed re-derivation covers independently. The degree walls bound a different quantity from the [[Statement:plane-gcd-bounds|gcd constraints]].
+
+<references />
+
Revision 272 as rendered
Moh analysed how the Puiseux roots at infinity of a hypothetical Jacobian pair can be configured. He then ran a computer search over degree pairs up to 100, which left six possible cases, and eliminated those separately.[1] The result is that the plane conjecture holds for every Jacobian pair of degree at most 100.
The proof has a documented history, and the atlas records all of it.
- 2005. L.-C. Wang recoded the algorithm, found exceptions beyond Moh's six cases, and modified it with approximate-root theory “to fulfill his claim”.[2]
- 2016–2022. Xu finds an unhandled split of minor roots and suggests that the case (99, 66) is open (unrefereed).[3]
- 2022. Guccione, Guccione, Horruitiner and Valqui, with their own case analysis, discard every pair with maximum degree below 125 except (72, 108). They thereby confirm Moh's bound and raise it to 108 (unrefereed).[4]
- 2025. T. Nguyen proves the complex case up to degree 104. The proof takes Moh's bound as input, together with the known gcd results.[5]
So the refereed bound is 104, the claimed bound is 108, and one case of the original argument is disputed in an unrefereed preprint. The disputed case lies inside the range that the unrefereed re-derivation covers independently. The degree walls bound a different quantity from the gcd constraints.
- ↑ T. T. Moh, On the Jacobian conjecture and the configurations of roots, J. reine angew. Math. 340 (1983) 140–212, doi:10.1515/crll.1983.340.140.
- ↑ L.-C. Wang, On the Jacobian conjecture, Taiwanese J. Math. 9 (2005) 421–431, doi:10.11650/twjm/1500407850.
- ↑ Y. Xu, Intersection Numbers and Split of Minor Roots, arXiv:1604.07683 (v4 2022).
- ↑ J. A. Guccione, J. J. Guccione, R. Horruitiner, C. Valqui, Increasing the degree of a possible counterexample to the Jacobian Conjecture from 100 to 108, arXiv:2204.14178.
- ↑ T. Nguyen, Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104, Quaest. Math. 48 (2025) 1291–1305, doi:10.2989/16073606.2025.2482655.