Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The purpose of this thesis was to examine the Hilbert Nullstellensatz and some of its more important corollaries. We approached the problem, working primarily with varieties and ideals. The major result is that the ideal associated with the variety of any ideal is the radical of that ideal. Definitions of the terms may be found in the text. It was necessary to develop considerable machinery in order to accomplish the proof. The preliminary details are done at a leisurely pace, with some propositions brought in for their own interest, rather than any application to our future proof. Propositions, which we consider only tools, will be labelled lemmas. Propositions, which are particularly important as independent results, are labelled theorems. Every effort was made to have the paper itself contained, although some experience with field extensions and polynomial rings is presupposed. / 2031-01-01

No description available.

No description available.

Aquest treball és una contribució a l'estudi de diferents problemes que sorgeixen de dues àrees fortament connexes de la Teoria de Grafs: etiquetaments i descomposicions. Molts etiquetaments de grafs deuen el seu origen als presentats l'any 1967 per Rosa. Un d'aquests etiquetaments, àmpliament conegut com a etiquetament graceful, va ser definit originalment com a eina per atacar la conjectura de Ringel, la qual diu que el graf complet d'ordre 2m+1 pot ser descompost en m copies d'un arbre donat de mida m. Aquí, estudiem etiquetaments relacionats que ens donen certes aproximacions a la conjectura de Ringel, així com també a una altra conjectura de Graham i Häggkvist que, en una forma dèbil, demana la descomposició d'un graf bipartit complet per un arbre donat de mida apropiada. Les principals contribucions que hem fet en aquest tema són la prova de la darrera conjectura per grafs bipartits complets del doble de mida essent descompostos per arbres de gran creixement i un nombre primer d'arestes, i la prova del fet que cada arbre és un subarbre gran de dos arbres pels quals les dues conjectures es compleixen respectivament. Aquests resultats estan principalment basats en una aplicació del mètode polinomial d'Alon. Un altre tipus d'etiquetaments, els etiquetaments magic, també són tractats aquí. Motivats per la noció de quadrats màgics de Teoria de Nombres, en aquest tipus d'etiquetaments volem asignar nombres enters a parts del graf (vèrtexs, arestes, o vèrtexs i arestes) de manera que la suma de les etiquetes assignades a certes subestructures del graf sigui constant. Desenvolupem tècniques basades en particions de certs conjunts d'enters amb algunes condicions additives per construir etiquetaments cycle-magic, un nou tipus d'etiquetament introduït en aquest treball i que estén la noció clàssica d'etiquetament magic. Els etiquetaments magic no donen cap descomposició de grafs, però les tècniques usades per obtenir-los estan al nucli d'un altre problema de descomposició, l'ascending subgraph decomposition (ASD). Alavi, Boals, Chartrand, Erdös i Oellerman, van conjecturar l'any 1987 que tot graf té un ASD. Aquí, estudiem l'ASD per grafs bipartits, una classe de grafs per la qual la conjectura encara no ha estat provada. Donem una condició necessària i una de suficient sobre la seqüència de graus d'un estable del graf bipartit de manera que admeti un ASD en que cada factor sigui un star forest. Les tècniques utilitzades estan basades en l'existència de branca-acoloriments en multigrafs bipartits. També tractem amb el sumset partition problem, motivat per la conjectura ASD, que demana una partició de [n] de manera que la suma dels elements de cada part sigui igual a un valor prescrit. Aquí donem la millor condició possible per la versió modular del problema que ens permet provar els millors resultats ja coneguts en el cas enter per n primer. La prova està de nou basada en el mètode polinomial. / This work is a contribution to the study of various problems that arise from two strongly connected areas of the Graph Theory: graph labelings and graph decompositions. Most graph labelings trace their origins to the ones presented in 1967 by Rosa. One of these labelings, widely known as the graceful labeling, originated as a means of attacking the conjecture of Ringel, which states that the complete graph of order 2m+1 can be decomposed into m copies of a given tree of size m. Here, we study related labelings that give some approaches to Ringel's conjecture, as well as to another conjecture by Graham and Häggkvist that, in a weak form, asks for the decomposition of a complete bipartite graph by a given tree of appropriate size. Our main contributions in this topic are the proof of the latter conjecture for double sized complete bipartite graphs being decomposed by trees with large growth and prime number of edges, and the proof of the fact that every tree is a large subtree of two trees for which both conjectures hold respectively. These results are mainly based on a novel application of the so-called polynomial method by Alon. Another kind of labelings, the magic labelings, are also treated. Motivated by the notion of magic squares in Number Theory, in these type of labelings we want to assign integers to the parts of a graph (vertices, edges, or vertices and edges) in such a way that the sums of the labels assigned to certain substructures of the graph remain constant. We develop techniques based on partitions of certain sets of integers with some additive conditions to construct cycle-magic labelings, a new brand introduced in this work that extends the classical magic labelings. Magic labelings do not provide any graph decomposition, but the techniques that we use to obtain them are the core of another decomposition problem, the ascending subgraph decomposition (ASD). In 1987, was conjectured by Alavi, Boals. Chartrand, Erdös and Oellerman that every graph has an ASD. Here, we study ASD of bipartite graphs, a class of graphs for which the conjecture has not been shown to hold. We give a necessary and a sufficient condition on the one sided degree sequence of a bipartite graph in order that it admits an ASD by star forests. Here the techniques are based on the existence of edge-colorings in bipartite multigraphs. Motivated by the ASD conjecture we also deal with the sumset partition problem, which asks for a partition of [n] in such a way that the sum of the elements of each part is equal to a prescribed value. We give a best possible condition for the modular version of the sumset partition problem that allows us to prove the best known results in the integer case for n a prime. The proof is again based on the polynomial method.

No description available.

NP search problems and reductions among them Renáta Ševčíková In the thesis we study the class of Total NP search problems. More attention is devoted to study the subclasses of Total NP search problems and reductions among them. We combine some known methods: the search trees and their relation to re- ductions, the Nullstellensatz refutation and the degree lower bound based on design to show that two classes of relativized NP search problems based on Mod-p counting principle and Mod-q counting principle, where p and q are different primes, are not reducible to each other. This thesis is finished by a new separation result for p = 2 and q = 3.

<p>This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of N_q(g) is now known.</p><p>At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.</p><p> </p>

This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known. At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.

La notion de T-radical d’un idéal permet à G.Cherlin de démontrer un Nullstellensatz dans les théories inductives d’anneaux. Nous proposons une analyse modèle-théorique de phénomènes connexes. En premier lieu, une réciproque de ce théorème nous conduit à une caractérisation des corps algébriquement clos, suggérant une version “positive” du travail de Cherlin, la théorie des idéaux T-radiciels. Ceux-ci se caractérisent par un théorème de représentation et sont associés à un théorème des zéros “positif”. Ces résultats se généralisent à la logique du premier ordre : grâce à la notion de classe spéciale, nous développons ensuite une théorie logique des idéaux. On peut encore parler d’idéaux premiers et radiciels, relativement à une classe de structures. Dans ce cadre, le théorème de représentation est une propriété intrinsèque des classes spéciales et le théorème des zéros une propriété de préservation logique, que nous appelons “complétude géométrique” et qui entretient des rapports étroits avec la modèle-complétude positive. Les algèbres basées en groupes de P.Higgins permettent d’appliquer ces résultats aux théories modèle-complètes de corps avec opérateurs additionnels. Dans certains cas “noethériens”, l’algèbre de coordonnées est un invariant algébrique des “variétés affines”. Enfin, il est possible à partir d’un ensemble de formules E de généraliser les classes spéciales et autres classes de structures. Notre théorie des idéaux logiques est de plus un cas particulier du phénomène de localisation étudié par M.Coste ; dans certaines situations, un bon choix de formules permet d’identifier les types complets d’une “algèbre” à des types de localisation / The notion of T-radical of an ideal allows G.Cherlin to prove a Nullstellensatz for inductive ring theories.We present here a model-theoretic analysis of closely related phenomena. At first, a reverse of this theorem leeds us to a characterization of algebraically closed fields, suggesting a “positive” version of Cherlin’s work, the theory of T-radical ideals. These are characterized by a representation theorem and associated to a “positive” Nullstellensatz. Those results are generalized to first order logic : thanks to the notion of special class, we then develop a logical theory of ideals. One may still speak about prime and radical ideals, relatively to a class of structures. In this setting, the representation theorem is an intrinsic property of special classes and the Nullstellensatz a logical preservation property, which we call “geometric completeness” and which is closely linked to positive model-completeness. The group-based algebras of P.Higgins allow us to apply these results to model-complete theories of fields with additional operators. In certain “noetherian” cases, the coordinate algebra is an algebraic invariant of “affine algebraic sets”. At last, it is possible from a set of formulas E to generalize special and other classes of structures. Moreover, our theory of logical ideals is a particular case of the localisation phenomenon studied by M.Coste ; in certain situations, a good choice of formulasleeds to an identification of the complete types of a given “algebra” with some localisation types