
Algorithms for the computation of Galois groups
Kubát, David ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
This thesis covers the topic of the computation of Galois groups over the rationals. Beginning with the classic algorithm by R. Stauduhar, we then review the theory necessary to explain the modular algorithm by K. Yokoyama. More precisely, we discuss the notion of the universal splitting ring of a polynomial. For a separable polynomial, we then study idempotents in the universal splitting ring. The modular algorithm involves computations in the ring of padic integers. Examples are given for polynomials of degree 3 and 4.


Discrete linear dynamical systems with control
Procházková, Zuzana ; Tůma, Jiří (advisor) ; Růžička, Pavel (referee)
Discrete linear dynamical systems with control Author: Zuzana Procházková Department: Department of Algebra Supervisor: doc. RNDr. Jiří Tůma, DrSc., Department of Algebra Abstract: In this thesis we describe elementary property of discrete linear dyna mical system. We define discrete linear dynamical system with control and its controllability and then we define descrete linear dynamical system with output and its observability. After that we show the duality of observability and con trollability with definition of dual system and its description. There are three problems solved in the last chapter. 1


Plane geometry problems solved by algebraic geometry
Trummová, Ivana ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis I focus on a certain part of algebraic geometry which studies plane curves and their intersection points. The main part is a proof of Bézout's theorem and an overview of its corollaries, which have an interesting geometric visualization. The most important corollary is the proof of associativity of adding points on elliptic curves. This fact is widely used in modern cryptography. 21


Coordinate Systems for GPS
Žváčková, Magdaléna ; Tůma, Jiří (advisor) ; Růžička, Pavel (referee)
V této práci se zabýváme souřadnicemi, které získáváme z GPS, běžně pou žívanými geodetickými souřadnicemi a tím, jak je mezi sebou převést. Nejprve jsou nalezeny vzorce pro převod z geodetických do kartézských a ty jsou poté řešeny jako soustava rovnic. Na to je použito několik numerických metod. Na zá kladě toho získáváme návod na to, jak převádět souřadnicové systémy mezi sebou. Dále se v krátkosti seznámíme s principy GPS, jak můžeme Zemi aproximovat a nakonec, jak převést povrch Země do mapy. 1


Using algebra in geometry
Paták, Pavel ; Růžička, Pavel (advisor) ; Šmíd, Dalibor (referee) ; Blagojevic, Pavle (referee)
Using algebra in geometry Pavel Paták Department: Department of Algebra Supervisor: Mgr. Pavel Růžička, Ph.D., Department of Algebra 1 Abstract In this thesis, we develop a technique that combines algebra, algebraic topology and combinatorial arguments and provides nonembeddability results. The novelty of our approach is to examine non embeddability arguments from a homological point of view. We illustrate its strength by proving two interesting theorems. The first one states that kdimensional skeleton of b 2k+2 k + k + 3 dimensional simplex does not embed into any 2kdimensional manifold M with Betti number βk(M; Z2) ≤ b. It is the first finite upper bound for Kühnel's conjecture of nonembeddability of simplices into manifolds. The second one is a very general topological Helly type theorem for sets in Rd : There exists a function h(b, d) such that the following holds. If F is a finite family of sets in Rd such that ˜βi ( G; Z2) ≤ b for any G F and every 0 ≤ i ≤ d/2 − 1, then F has Helly number at most h(b, d). If we are only interested whether the Helly numbers are bounded or not, the theorem subsumes a broad class of Helly types theorems for sets in Rd . Keywords: Homological Nonembeddability, Helly Type Theorem, Kühnel's conjecture of nonembeddability of ske leta of simplices into manifolds


The arity of NU polymorphisms
Draganov, Ondřej ; Barto, Libor (advisor) ; Růžička, Pavel (referee)
This paper deals with an arity of NU polymorphisms of relational structures. The goal is to simplify and clearly describe an already existing example of a relational structure, which has an NU polymorphism, but no NU polymorphisms of low arity in respect to arity of relations and to a number of elements in the relational structure. We explicitly describe mary relational structures with n elements, n ≥ 2, m ≥ 3, which have no NU polymorphisms of arity (m − 1)2n−2 , but have an NU polymorphism of arity (m − 1)2n−2 + 1, which is constructed in the paper, and binary relational structures with n elements, n ≥ 3, which have no NU polymorphisms of arity 22n−3 , but have an NU polymorphism of arity 22n−3 + 1.


Relational Approach to Universal Algebra
Opršal, Jakub ; Barto, Libor (advisor) ; Růžička, Pavel (referee) ; Mayr, Peter (referee)
Title: Relational Approach to Universal Algebra Author: Jakub Opršal Department: Department of Algebra Supervisor: doc. Libor Barto, Ph.D., Department of Algebra Abstract: We give some descriptions of certain algebraic properties using rela tions and relational structures. In the first part, we focus on Neumann's lattice of interpretability types of varieties. First, we prove a characterization of vari eties defined by linear identities, and we prove that some conditions cannot be characterized by linear identities. Next, we provide a partial result on Taylor's modularity conjecture, and we discuss several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and the analogue for idempotent va rieties with a cube term. In the second part, we give a relational description of higher commutator operators, which were introduced by Bulatov, in varieties with a Mal'cev term. Furthermore, we use this result to prove that for every algebra with a Mal'cev term there exists a largest clone containing the Mal'cev operation and having the same congruence lattice and the same higher commu tator operators as the original algebra, and to describe explicit (though infinite) set of identities describing supernilpotence...


Problém realizace von Neumannovsky regulárních okruhů
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: The realization problem for von Neumann regular rings Author: Samuel Mokriš Department: Department of Algebra Supervisor of the master thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: With every unital ring R, one can associate the abelian monoid V (R) of isomor phism classes of finitely generated projective right Rmodules. Said monoid is a conical monoid with orderunit. Moreover, for von Neumann regular rings, it satisfies the Riesz refinement property. In the thesis, we deal with the question, under what conditions an abelian conical re finement monoid with orderunit can be realized as V (R) for some unital von Neumann regular ring or algebra, with emphasis on countable monoids. Two generalizations of the construction of V (R) to the context of nonunital rings are presented and their interrelation is analyzed. To that end, necessary properties of rings with local units and modules over such rings are devel oped. Further, the construction of Leavitt path algebras over quivers is presented, as well as the construction of a monoid associated with a quiver that is isomorphic to V (R) of the Leavitt path algebra over the same quiver. These methods are then used to realize directed unions of finitely generated free abelian monoids as V (R) of algebras over any given field. A method...


Multivariate cryptography
Jančaříková, Irena ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
This thesis deals with multivariate cryptography. It includes specifically a description of the MQ problem and the proof of it's NPcompletness. In the part of the MQ problem there is a description of a general pattern for the creation of the public part of asymetric cryptosystems based on the MQ problem. It this part the thesis describes the QMLE problem, which is important for the figure of the cryptosystem private key based on the MQ problem. Further, the thesis includes a description of the influence of the structure display, which appears in the QMLE problem, on time solution complexity of QMLE problem. The influence of time complexity has been detected by means of experimental measurement with programed algorithm. At the end of the thesis there is specified description of selected multivariety cryptosystems based on the MQ problem. Selected cryptosystems are provided with detailed description of encryption and decryption by means of selected cryptosystems and time estimations of these operations. The thesis includes estimations of memory requirements on saving of private and public key of the selected cryptosystems. Powered by TCPDF (www.tcpdf.org)


An algorithmic approach to resolutions in representation theory
Ivánek, Adam ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we describe an algorithm and implement a construction of a projective resolution and minimal projective resolution in the representation the ory of finitedimensional algebras. In this thesis finitedimensional algebras are KQ /I where KQ is a path algebra and I is an admissible ideal. To implement the algorithm we use the package QPA [9] for GAP [2]. We use the theory of Gröbners basis of KQmodules and the theory described in article Minimal Pro jective Resolutions written by Green, Solberg a Zacharia [5]. First step is find a direct sum such that i∈Tn fn∗ i KQ = i∈Tn−1 fn−1 i KQ ∩ i∈Tn−2 fn−2 i I. Next important step to construct the minimal projective resolution is separate nontri vial Klinear combinations in i∈Tn−1 fn−1 i I + i∈Tn fn i J from fn∗ i . The Modules of the minimal projective elements are i∈Tn (fn i KQ)/(fn i I). 1
