Algebrák, varietások és algoritmusok = Algebras, varieties and algorithms

A pályázat keretében három fő témakörben --- általános algebra és alkalmazásai, félcsoportelmélet és döntési problémák bonyolultsága --- nyertünk eredményeket. A kutatás jelentős része hazai, illetve külföldi kutatókkal való együttműködésben született. Kiterjesztettük Rosenberg teljességi tételét a Slupecki-klón környezetében, általánosítottuk Wiegold dichotómiatételét parallelogramma-algebrákra, és az eddig ismerteknél egyszerűbb jellemzést adtunk a kongruenciamodularitásra. Karakterizáltuk a döntések kvalitatív modellezésére szolgáló hasznossági függvényeket, és eljárást adtunk egyszerűbb függvények kompozíciójaként való előállításukra. Kiterjesztettük a McAlister-Lawson-féle elmélet egyes fedési, illetve beágyazási tételeit a lokálisan inverz félcsoportokra a majdnem faktorizálhatóság általánosításával, illetve a bal és kétoldali megszorításos félcsoportokra a W-szorzatba való beágyazhatóság jellemzésével. A homomorfizmus-problémára vonatkozó dichotómiasejtést bebizonyítottuk két különböző új algebraosztályra. Megmutattuk, hogy egy véges idempotens algebra pontosan akkor örökletesen véges relációbázisú, ha van élkifejezése. Széles algebraosztályok esetén algebrailag jellemeztük az egyenletrendszer-problémák komplexitását. Algebrai eszközökkel bizonyítottuk Valeriote egy sejtését a reflexív irányított gráfok speciális esetére, valamint igazoltuk Stahl Kneser-gráfokra vonatkozó sejtésének speciális esetét. | The results of the project belong to three areas: universal algebra and applications, semigroup theory, and complexity theory. Most of the research was carried out in international cooperation. We extended Rosenberg’s completeness theorem in the neighborhood of Slupecki’s clone, we generalized Wiegold’s dichotomy theorem to parallelogram algebras, and we found a characterization for congruence modularity which is simpler than the known criteria. We characterized utility functions which provide a tool for qualitative modelling of decision-making, and gave a procedure to express them as compositions of simpler functions. We extended some of the covering and embedding theorems of the McAlister-Lawson theory for locally inverse semigroups by generalizing almost factorizability, and for left and two-sided restriction semigroups by characterizing embeddability in W-products. We verified the constraint satisfaction problem dichotomy conjecture for two new classes of algebras. We proved that a finite idempotent algebra is inherently finitely related if and only if it has an edge term. We gave an algebraic characterization of the complexity of the problems of systems of equations for broad classes of finite algebras. Based on algebraic methods, we confirmed the Valeriote conjecture for the special case of finite reflexive digraphs, and verified a special case of the Stahl conjecture on Kneser graphs

A unified approach to polynomial sequences with only real zeros

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for certain classes of graphs, while showing that they are false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function $f(z)$ by a sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by $f(z)$. Compared to the usual Pad\'e approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Pad\'e Conjecture and Nuttall's Conjecture for the sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ in the complement \mathbb{CP}^1\setminus \D_{f}, where \D_{f} is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family $\{r_{n}(z)\}_{n\in \mathbb{N}}$. As an application we settle the so-called 3-conjecture of Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.Comment: 25 pages, 8 figures, LaTeX2e, revised version to appear in Advances in Mathematic

Stahl's Theorem (aka BMV Conjecture): Insights and Intuition on its Proof

The Bessis-Moussa-Villani conjecture states that the trace of $\exp(A-tB)$ is, as a function of the real variable $t$, the Laplace transform of a positive measure, where $A$ and $B$ are respectively a hermitian and positive semi-definite matrix. The long standing conjecture was recently proved by Stahl and streamlined by Eremenko. We report on a more concise yet self-contained version of the proof.Comment: Conference pape