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Vortragende Person/Vortragende Personen:
Alejandro Gonzalez Nevado

We develop a two-step approach to improve known inner bounds on extreme roots of sequences of univariate real-rooted polynomials coming from combinatorics. The process involves the injection of the chosen univariate sequence of real-rooted polynomials into a multivariate sequence of real-zero polynomials through a finer counting procedure over the combinatorial objects attached to the univariate polynomials in the initial sequence. Such injection happens through an extension that is constructed differently for each combinatorial feature we are studying and is directly connected with the combinatorial structure of the object in a non-trivial non-standardizable way. In particular, we can show that the obvious general multivariate extensions are of no help in improving the bounds through this method. The second ingredient of this process consists in the use of a (relatively easy to compute) relaxation for rigidly convex sets of real-zero polynomials whose justification as a relaxation comes from the Helton-Vinnikov theorem when more than two variables are at play. Through the very univariate rudiments of this approach, we solve a problem of Mező on the (first-order) asymptotic growth of extreme roots of Eulerian polynomials. Furthermore, we prove that, using the whole power of the multivariate setting, our method provides inner bounds closer to the extreme roots in the second-order asymptotics. Remarkably, we can compute these better bounds without even knowing the full polynomials. The example of Eulerian polynomials opens the door to the study of improvements in the known inner bounds of extreme roots of other sequences of combinatorial polynomials. Therefore, in a broad sense, this behavior suggests the possibility of implementing a mathematical program (Mindelsee Program) oriented towards the search for finer ways of counting features in combinatorial objects using real-zero multivariate (or even non-commutative) polynomials as carriers of such additional information. Finally, as this example shows, these real-zero multivariate extensions can provide further information about the extreme roots of the univariate versions of these polynomials.