斯梅爾問題

斯梅爾問題（英語：Smale's problems）是美國數學家斯蒂芬·斯梅爾於1998年提出的18個當時未解決的數學問題。^[1]^[2]時任國際數學聯盟副主席、俄國數學家弗拉基米爾·阿諾爾德當時參照20世紀初希爾伯特的23個問題而向世界上的主要數學家徵集面向21世紀的數學問題，斯梅爾便是在此背景下提出了斯梅爾問題，作為對阿諾爾德的答覆。

斯梅爾的18個問題

#	問題	進展
1	黎曼猜想	未解決，參見希爾伯特第八問題
2	龐加萊猜想	2003年，格里戈里·佩雷爾曼證明完成^[3]^[4]^[5]
3	P=NP是否成立？	未解決
4	單變量多項式的整數零點^[6]^[7]	未解決
5	丟番圖曲線高度的界	未解決
6	天體力學中相對平衡狀態數量的有限性	2012年，A. Albouy與V. Kaloshin證明五個天體相對平衡狀態^[8]
7	二維球面上點的分布	未解決
8	將動力學引入經濟學理論	未解決
9	找出一個強多項式時間的線性規劃解法	未解決
10	封閉引理（英語：Pugh's closing lemma）	M. Asaoka、K. Irie部分解決^[9]
11	一維動力系統一般是否為雙曲型？	Kozlovski、Shen、van Strien部分解決^[10]
12	微分同胚的中心化子	Christian Bonatti、Sylvain Crovisier、Amie Wilkinson部分解決^[11]
13	希爾伯特第十六問題	未解決
14	洛倫茲吸引子	2001年，沃里克·塔克爾（英語：Warwick Tucker）（Warwick Tucker）證明完成^[12]
15	納維-斯托克斯方程	未解決
16	雅可比猜想	未解決
17	解多項式方程組	已解決，C. Beltrán、L. M. Pardo、F. Cucker、P. Bürgisser、P. Lairez各自完成部分證明^[13]^[14]^[15]^[16]^[17]
18	智能的極限^[18]	未解決

參見

千禧年大獎難題

參考文獻

^ Smale, Steve. Mathematical Problems for the Next Century. Mathematical Intelligencer. 1998, 20 (2): 7–15doi=10.1007/bf03025291.
^ Smale, Steve. Mathematical problems for the next century. Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (編). Mathematics: frontiers and perspectives. American Mathematical Society. 1999: 271–294. ISBN 0821820702.
^ Perelman, Grigori. The entropy formula for the Ricci flow and its geometric applications. 2002. arXiv:math.DG/0211159  |class=被忽略 (幫助).
^ Perelman, Grigori. Ricci flow with surgery on three-manifolds. 2003. arXiv:math.DG/0303109  |class=被忽略 (幫助).
^ Perelman, Grigori. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. 2003. arXiv:math.DG/0307245  |class=被忽略 (幫助).
^ Shub, Michael; Smale, Steve. On the intractability of Hilbert’s Nullstellensatz and an algebraic version of "NP≠P?". Duke Math. J. 1995, 81: 47–54. Zbl 0882.03040. doi:10.1215/S0012-7094-95-08105-8.
^ Bürgisser, Peter. Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics 7. Berlin: Springer-Verlag. 2000: 141. ISBN 3-540-66752-0. Zbl 0948.68082.
^ Albouy, A.; Kaloshin, V. Finiteness of central configurations of five bodies in the plane. Annals of Mathematics. 2012, 176: 535–588. doi:10.4007/annals.2012.176.1.10.
^ Asaoka, M.; Irie, K. A $C \infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geometric and Functional Analysis. 2016, 26: 1245–1254. doi:10.1007/s00039-016-0386-3.
^ Kozlovski, O.; Shen, W.; van Strien, S. Density of hyperbolicity in dimension one. Annals of Mathematics. 2007, 166: 145–182. doi:10.4007/annals.2007.166.145.
^ Bonatti, C.; Crovisier, S.; Wilkinson, A. The C¹-generic diffeomorphism has trivial centralizer. Publications Mathématiques de l'IHÉS. 2009, 109: 185–244. arXiv:0804.1416  . doi:10.1007/s10240-009-0021-z.
^ Tucker, Warwick. A Rigorous ODE Solver and Smale's 14th Problem (PDF). Foundations of Computational Mathematics. 2002, 2 (1): 53–117. doi:10.1007/s002080010018.
^ Beltrán, Carlos; Pardo, Luis Miguel. On Smale's 17th Problem: A Probabilistic Positive answer (PDF). Foundations of Computational Mathematics. 2008, 8 (1): 1–43 [2018-09-30]. doi:10.1007/s10208-005-0211-0. （原始內容存檔 (PDF)於2020-06-07）.
^ Beltrán, Carlos; Pardo, Luis Miguel. Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions (PDF). Journal of the American Mathematical Society. 2009, 22: 363–385 [2018-09-30]. Bibcode:2009JAMS...22..363B. doi:10.1090/s0894-0347-08-00630-9. （原始內容存檔 (PDF)於2020-06-07）.
^ Cucker, Felipe; Bürgisser, Peter. On a problem posed by Steve Smale. Annals of Mathematics. 2011, 174 (3): 1785–1836. doi:10.4007/annals.2011.174.3.8.
^ Lairez, Pierre. A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time. Foundations of Computational Mathematics. 2016,. to appear.
^ Shub, Michael; Smale, Stephen. Complexity of Bézout's theorem. I. Geometric aspects. J. Amer. Math. Soc. 1993, 6 (2): 459–501. doi:10.2307/2152805. .
^ http://recursed.blogspot.jp/2006/02/tucson-day-3-interview-with-steve.html （頁面存檔備份，存於網際網路檔案館） Friday, February 03, 2006 Tucson - Day 3 - Interview with Steve Smale

[1] Smale, Steve. Mathematical Problems for the Next Century. Mathematical Intelligencer. 1998, 20 (2): 7–15doi=10.1007/bf03025291.

[2] Smale, Steve. Mathematical problems for the next century. Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (編). Mathematics: frontiers and perspectives. American Mathematical Society. 1999: 271–294. ISBN 0821820702.

[3] Perelman, Grigori. The entropy formula for the Ricci flow and its geometric applications. 2002. arXiv:math.DG/0211159  |class=被忽略 (幫助).

[4] Perelman, Grigori. Ricci flow with surgery on three-manifolds. 2003. arXiv:math.DG/0303109  |class=被忽略 (幫助).

[5] Perelman, Grigori. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. 2003. arXiv:math.DG/0307245  |class=被忽略 (幫助).

[6] Shub, Michael; Smale, Steve. On the intractability of Hilbert’s Nullstellensatz and an algebraic version of "NP≠P?". Duke Math. J. 1995, 81: 47–54. Zbl 0882.03040. doi:10.1215/S0012-7094-95-08105-8.

[7] Bürgisser, Peter. Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics 7. Berlin: Springer-Verlag. 2000: 141. ISBN 3-540-66752-0. Zbl 0948.68082.

[8] Albouy, A.; Kaloshin, V. Finiteness of central configurations of five bodies in the plane. Annals of Mathematics. 2012, 176: 535–588. doi:10.4007/annals.2012.176.1.10.

[9] Asaoka, M.; Irie, K. A $C \infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geometric and Functional Analysis. 2016, 26: 1245–1254. doi:10.1007/s00039-016-0386-3.

[10] Kozlovski, O.; Shen, W.; van Strien, S. Density of hyperbolicity in dimension one. Annals of Mathematics. 2007, 166: 145–182. doi:10.4007/annals.2007.166.145.

[11] Bonatti, C.; Crovisier, S.; Wilkinson, A. The C¹-generic diffeomorphism has trivial centralizer. Publications Mathématiques de l'IHÉS. 2009, 109: 185–244. arXiv:0804.1416  . doi:10.1007/s10240-009-0021-z.

[12] Tucker, Warwick. A Rigorous ODE Solver and Smale's 14th Problem (PDF). Foundations of Computational Mathematics. 2002, 2 (1): 53–117. doi:10.1007/s002080010018.

[13] Beltrán, Carlos; Pardo, Luis Miguel. On Smale's 17th Problem: A Probabilistic Positive answer (PDF). Foundations of Computational Mathematics. 2008, 8 (1): 1–43 [2018-09-30]. doi:10.1007/s10208-005-0211-0. （原始內容存檔 (PDF)於2020-06-07）.

[14] Beltrán, Carlos; Pardo, Luis Miguel. Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions (PDF). Journal of the American Mathematical Society. 2009, 22: 363–385 [2018-09-30]. Bibcode:2009JAMS...22..363B. doi:10.1090/s0894-0347-08-00630-9. （原始內容存檔 (PDF)於2020-06-07）.

[15] Cucker, Felipe; Bürgisser, Peter. On a problem posed by Steve Smale. Annals of Mathematics. 2011, 174 (3): 1785–1836. doi:10.4007/annals.2011.174.3.8.

[16] Lairez, Pierre. A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time. Foundations of Computational Mathematics. 2016,. to appear.

[17] Shub, Michael; Smale, Stephen. Complexity of Bézout's theorem. I. Geometric aspects. J. Amer. Math. Soc. 1993, 6 (2): 459–501. doi:10.2307/2152805. .

[18] ttp://recursed.blogspot.jp/2006/02/tucson-day-3-interview-with-steve.html （頁面存檔備份，存於網際網路檔案館） Friday, February 03, 2006 Tucson - Day 3 - Interview with Steve Smale

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