The van der waerden conjecture: two proofs in one year

Abstract

One of the famous open problems in combinatorial theory was the van der Waerden conjecture on permanents of doubly stochastic matrices. After more than fifty years in which it managed to resist attacks it has finally been proved. As so often happens two mathematicians gave independent proofs nearly at the same time. At the end of 1980 the news that G. P. Egoritsjev (F. H. EropbiyeB) [2] had found a proof spread quickly and within a few months translations and expositions of the proof were circulating (cf. D. E. Knuth [5], J. H. van Lint [6]). It came as quite a shock when Matemati~eski Zametki 29 No. 6 appeared a few months ago with a paper by D. I. Falikman ()I. I/I. ~a~lnI<Man) [3], submitted 14.5. 1979 (!), with a completely different proof of the conjecture. Perhaps even more surprising is the fact that the two proofs have as a common feature that they use an elegant inequality, due to A. D. Alexandroff and W. Fenchel, which until quite recently seemed to be unknown to combinatorialists. From the literature it is obvious that geometers certainly know about the inequality. In this note we shall describe and compare the two new proofs and the ideas that led to their discovery. Let A be a square matrix of size n with entries a 6 (1 <~ i, j ~< n). We define the permanent of A (notation: perA) by