Page 12 - Textos de Matemática Vol. 44

2 A. KOVACˇEC
for essentially a chronological survey of a selection of typical papers. To al- low the potential reader rapid identification we begin the report of each paper in a form like [XY89: numerical range, determinantal range] where XY89 is a mnemonic for the author’s family name(s) and the publication year and the following terms are key words of topics treated in it. Within such an expo- sition, some related papers of her’s are also mentioned. For reasons of space, the bibliographic entries mention only three works of which Nat´alia is not co- author. These contain substantial background to numerical ranges and matrix inequalities.
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This paper was written with some care and reflects the facts to the best of author’s knowledge. But time and energy constraints impeded an in depth study of all the material here referred. And only such an analysis can, as every mathematician and historian will know, assure that the concepts are histori- cally and technically correct, and not distorted under compactification. Ideally a report as this one would even straighten out slips and explanatory short- comings in the original papers (in case there are any). Maybe we have done so haphazardly at times; but more modestly our aim is to inform with somewhat more coherence and detail than in mathematical review organs about some papers central to Nat´alia’s research; we do not claim to be a reliable source in all the details.
May some readers of this report, for the benefit of linear algebra, be driven to collaborate with Nat´alia and tap into her wide expertise!
1. Permanents
[B82]: Apparently the first paper recorded in Zentralblatt is a 1979 paper on the evaluation of permanents which was later published in extended form in the renowned Pacific Journal of Mathematics [B82]. The fact that more than half of the pages in [B85] is dedicated to the van der Waerden conjecture of 1926 which says that the permanent of an n × n doubly stochastic matrix is at least n!/nn, and the bibliography to the follow-up [B92] suggest that her interest in this topic might have been sparked by it. It was solved in 1981 by Falikman and Egorychev and was at that time very much in vogue. Another influence might have been the awareness of her father in law of the interest permanents have in statistical physics (although they where not called permanents there). In fact they are since decades also a hot topic in computer science as computationally hard to compute, and in combinatorics.
Let A = (aij) be an n×n matrix and let x be an n×1 column of variables. Then the matrix equation z = Ax produces n linear forms zi = zi(x1, . . . , xn),