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By Irmgard Süss (auth.), E. F. Beckenbach, Prof. Dr. Wolfgang Walter (eds.)

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Additional info for General Inequalities 3: 3rd International Conference on General Inequalities, Oberwolfach, April 26 – May 2, 1981

Example text

S E n* and n S I J, n then per(S) > This result, however, is an immediate consequence of Theorem 1 and a theorem of Marcus and Newman (Lemma 2). 5. EGORYCEV' S PROOF Egorycev's proof of the van der Waerden conjecture (Theorem 2) depends on a permanental inequality derived by Egorycev from an inequality for mixed discriminants of Alexandrov [1]. It also depends on results of Marcus and Newman and of London. Let ~,A 2 , permutation row is the ••• ,An be cr E Sn' ith discriminant of let n Xn A(cr) real symmetric matrices.

Since n is n Now ..... ""· as i ..... ) > sjrr(A. ) would tend to oo E l contradiction. Therefore A E n* and as i oo, which is a -+ n F (A) E = min{F E (s)/s E n*}. 0 n We are now ready for Falikman' s key theorem. THEOREM 3. point of F s ~- c = in Let n*. n E> 0 and let Then A = J • A = (a .. ) lJ E nn* be a minimum n We shall use the following abbreviated notation: s/IT(A), and oF E d .. lxij ~ =A ' i,j=l,2, ... ,n. Henryk Mine 32 Thus A by adding h to the entries is the matrix obtained from ~ in positions Ah so that (i,l).

N In 1979 Fried1and [14] used a method devised by Bang [2] to for all prove S E that per(S) ~ 1/en (7) for a11 s E n • Bang [3] also proved that n per ( s) (8) for any s E n . n ~ 1 n-1 1,e 28 Henryk Mine About the timethat Friedland's and Bang's results appeared (on May 14, 1979, to be exact), a Russian mathematician, D. I. Falikman, submitted his paper [9] to the Matematiceske Zametki. In this paper Falikman gave a complete proof of the van der Waerden permanent conjecture (Theorem 1; that is, the conjecture without the assertion on the uniqueness of the minimizing matrix).

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