Page 15 - Textos de Matemática Vol. 39

E. MARQUES DE SA´ AND THE INTERLACING INEQUALITIES 5
suggested to him a problem on invariant polynomials, he decided to study some polynomials that came up, call them h1, h2, . . . , hn, and he plotted the points with coordinates (i,degree of hi) for the relevant values of i (there were good reasons to do that). Then he joined these points and obtained, in all examples, polygonal lines with the concavity facing downwards! Tired of getting this in all cases, he tried to build examples where it didn’t happen. He couldn’t, so he suspected that the statement was true in general, and began looking for a proof. He entered the realm where the computer is useless and he did well! Very well indeed. One could say: it would have been harder with pencil and paper, and harder still with stylus and tablet.
The details and the reasons why and how those polynomials appear, and why their degrees play such an important role, can only be understood by studying the proofs. Marques de Sa´’s talent and deep originality in the treat- ment of convexity can only be appreciated by reading the paper. Whoever reads the papers by Marques de Sa´ [2] and Thompson [4] will also understand the importance of knowledge of different parts of Mathematics. Note Sa´’s and Thompson’s treatment of the crucial question of convexity. Thompson himself later wrote: “Thompson used a lengthy, rather brute force, technique to over- come the difficulty with the degrees, whereas Sa´ invented a clever argument involving (unbelievably) the approximation property of a Lebesgue integral by integrals of step functions. (. . . ) Later, another rather combinatorial proof cov- ering the same difficulty with degrees was obtained by P. Y. Chen. (. . . ) These proofs have little in common and all are hard to follow (. . . ) and it’s probably true that no one of the three authors understands the other two proofs”.
I agree with this last statement, which struck me so much that, in a Dublin talk in 1986, I added the following: “and it’s probably true that, excepting the three authors, no one understands at least one of the three proofs”.
Thompson was writing about the proof by Sa´, his student P. Y. Chen’s, and his own. Later still another one was found by Ion Zabalha [6, 7], and Marques de Sa´ also published a simplification of his original argument.
We celebrate today an important date in the life of Eduardo Marques de Sa´, and it’s a pleasure for me to associate myself to the event. I left here a summary of what I think are the most significant events of his curriculum, to pay tribute to him and — perhaps the greatest honour — to call the attention of young researchers to important open problems. I left out many facts worth noting: Marques de Sa´’s dedication to teaching, the training of other mathe- maticians, and other exploits in research. After interlacing, Marques de Sa´ did not stop and many original, quality results are due to him. I will not attempt to describe them, not because they do not deserve it but because space limita- tions led me to a choice dictated by my preferences. Last but not least, to be complete I would have to study all of Marques de S´a’s work, and I don’t think at my age I could do that. I leave it to others.