凡得瓦登猜想

线性代数中,Van der Waerden猜想是一个关于积和式的命题,其具体内容如下:

对于任意的双转移矩阵,其积和式的值大于等于

注意到该下界在所有元素均为时成立。该猜想由Bartel Leendert van der Waerden英语Bartel Leendert van der Waerden在1926年提出[1],在1980年由B. Gyires[2],1981年由G. P. Egorychev[3]和D. I. Falikman[4]独立证明,其中Egorychev的证明用到了Alexandrov–Fenchel不等式英语Alexandrov–Fenchel inequality[5]由于这项工作,Egorychev和Falikman赢得了1982年的Fulkerson奖英语Fulkerson Prize[6]

参考

  1. ^ van der Waerden, B. L., Aufgabe 45, Jber. Deutsch. Math.-Verein., 1926, 35: 117 .
  2. ^ Gyires, B., The common source of several inequalities concerning doubly stochastic matrices, Publicationes Mathematicae Institutum Mathematicum Universitatis Debreceniensis, 1980, 27 (3-4): 291–304, MR 0604006 .
  3. ^ Egoryčev, G. P., Reshenie problemy van-der-Vardena dlya permanentov, Krasnoyarsk: Akad. Nauk SSSR Sibirsk. Otdel. Inst. Fiz.: 12, 1980, MR 0602332 (俄语) . Egorychev, G. P., Proof of the van der Waerden conjecture for permanents, Akademiya Nauk SSSR, 1981, 22 (6): 65–71, 225, MR 0638007 (俄语) . Egorychev, G. P., The solution of van der Waerden's problem for permanents, Advances in Mathematics, 1981, 42 (3): 299–305, MR 0642395, doi:10.1016/0001-8708(81)90044-X .
  4. ^ Falikman, D. I., Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Akademiya Nauk Soyuza SSR, 1981, 29 (6): 931–938, 957, MR 0625097 (俄语) .
  5. ^ Brualdi (2006) p.487
  6. ^ Fulkerson Prize页面存档备份,存于互联网档案馆), Mathematical Optimization Society, retrieved 2012-08-19.