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雅可比猜想斷言，若一 polynomial map F:C^n->C^n 具有 nonzero constant Jacobian determinant，則其為 automorphism。 在平面上，該猜想最早由 Keller（1939）提出，其內容為：C[x,y]（複數上二變數多項式環）到自身的環映射 F，若固定 C 且將 x、y 分別映至 f、g，則該映射為自同構 iff 的充要條件是其雅可比行列式 f_xg_y-f_yg_x 為非零常數。該條件易證明為必要。

多年來已有至少五篇已發表的錯誤證明，以及許多不正確的嘗試。2004 年 11 月，Hochster（2004）發送電子郵件宣布 Carolyn Dean 提出新證明。然而，該證明亦存在錯誤。

JacobianConjectureCounterexample

2026 年 7 月，Alpöge（2026）宣布以下多項式反例，並將其歸功於 AI 系統 Fable。寫作 F=(P,Q,R)，其座標多項式為

其 Jacobian determinant 為非零常數

detJ_F=-2,

(4)

但下列三個相異點

(0,0,-1/4),(1,-3/2,(13)/2),(-1,3/2,(13)/2),

(5)

具有相同像，因為

F(0,0,-1/4)=F(1,-3/2,(13)/2)=F(-1,3/2,(13)/2)=(-1/4,0,0).

(6)

因此，雅可比猜想在維度 3 為偽，且藉由附加恆等座標，在所有維度 n>=3 亦為偽（Zhang 2026）。平面情形仍屬未解。

雅可比猜想為 Smale's problems 之一。

See also

Invertible Polynomial Map, Jacobian, Polynomial Map, Smale's Problems

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References

Abhyankar, S. S. Lectures on Expansion Techniques in Algebraic Geometry. Bombay, India: Tata Institute of Fundamental Research, 1977.Alpöge, L. X post, July 20, 2026. https://x.com/__alpoge__/status/2079028340955197566.Bass, H. "Conjecture jacobienne et opérateurs différentiels." Mém. Soc. Math. France, No. 38, 39-50, 1989.Bass, H.; Connell, E. H.; and Wright, D. "The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse." Bull. Amer. Math. Soc. 7, 287-330, 1982.Becker, T. and Weispfenning, V. Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, p. 330, 1993.Drużkowski, L. M. "The Jacobian Conjecture." IMPAN Preprint 492. Kraków, Poland: Math. Inst. Jagiellonian University, 1991.Formanek, E. "Observations About the Jacobian Conjecture." Houston J. Math. 20, 369-380, 1994.Hochster, M. "Lectures on Jacobian Conjecture." sci.math.research post forwarded by I. Algol. Nov. 11, 2004.Keller, O.-H. "Ganze Cremona Transformationen." Monatsh. für Math. u. Phys. 47, 299-306, 1939.Meisters, G. H. "Jacobian Problems in Differential Equations and Algebraic Geometry." Rocky Mountain J. Math. 12, 679-705, 1982.Meisters, G. H. "Wanted: A Bad Matrix." Amer. Math. Monthly 102, 546-550, 1995.Smale, S. "Mathematical Problems for the Next Century." Math. Intelligencer 20, No. 2, 7-15, 1998.Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Providence, RI: Amer. Math. Soc., 2000.Zhang, Z. "Direct Consequences of the Three-Dimensional Counterexample to the Jacobian Conjecture." July 20, 2026. https://zzhang-iu.github.io/papers/direct-consequences-jacobian/.

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Jacobian Conjecture

Cite this as:

Weisstein, Eric W. "Jacobian Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobianConjecture.html

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