Let p be a prime, p a power of p and 𝔽q the finite field with q elements. Any function φ: 𝔽q → 𝔽q can be unqiuely represented by a polynomial, 𝔽φ of degree < q. If the map x ↦ Fφ(x) induces a permutation on the underlying field we say Fφ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F(x) induce a permutation on 𝔽q. In the second chapter we are concerned the permutation behavior of the polynomial gn,q, a q-ary version of the reversed Dickson polynomial, when the integer n is of the form n = qa - qb - 1. This leads to the third chapter where we consider binomials and trinomials taking special forms. In this case we are able to give explicit conditions that guarantee the given binomial or trinomial is a permutation polynomial. In the fourth chapter we are concerned with permutation polynomials of 𝔽q, where q is even, that can be represented as the sum of a power function and a linearized polynomial. These types of permutation polynomials have applications in cryptography. Lastly, chapter five is concerned with a conjecture on monomial graphs that can be formulated in terms of polynomials over finite fields.

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In this report, we present some of the results achieved by investigating two-dimensional monomial cellular automata modulo m, where m is a non-zero positive integer. Throughout the experiments, we work with the von Neumann neighbourhood and apply the same local rule based on modular multiplication. The purpose of the study is to examine the behaviour of these cellular automata in three different environments, (i.e. the infinite plane, the finite plane and the torus), by means of elementary number theory. We notice how the distance between each pair of cells with state 0 influences the evolution of the automaton and the convergence of its configurations. Similar impact is perceived when the cells attain the values of Euler's-<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cphi" />function or of integers with common divisors with m, when m &gt; 2. Alongside with the states of the cells, the evolution of the automaton, as well as the convergence of its configurations, are also decided by the values attributed to m, whether it is a prime, a prime power or a multiple of primes and/or prime powers.

An artinian standard graded algebra has the weak Lefschetz property if the multiplication by a general linear form induces maps of maximal rank between consecutive degree components. It has the strong Lefschetz property if the multiplication by powers of a general linear form also induce maps of maximal rank between the appropriate degree components. These properties are mainly studied for the constraints they place, when present, on the Hilbert series of the algebra. While the majority of research on the Lefschetz properties has focused on characteristic zero, we primarily consider the presence of the properties in positive characteristic. We study the Lefschetz properties by considering the prime divisors of determinants of critical maps. First, we consider monomial complete intersections in a finite number of variables. We provide two complements to a result of Stanley. We next consider monomial almost complete intersections in three variables. We connect the characteristics in which the weak Lefschetz property fails with the prime divisors of the signed enumeration of lozenge tilings of a punctured hexagon. Last, we study how perturbations of a family of monomial algebras can change or preserve the presence of the Lefschetz properties. In particular, we introduce a new strategy for perturbations rooted in techniques from algebraic geometry.

We have investigated several monomial and permutation progenitors, including 2*8 : [8 : 2], 2*18 : [(22 x 3) : (3x2)], 2*16 : [22 : 4], and 2*16 : 24, 5*2 :m [4•22], 5*2 :m [(4x2) :• 2], 103∗2 :m [17 : 2] and 103∗4 :m [17 : 4]. We have discovered original, to the best of our knowledge, symmetric presentations of a number of finite groups, including PSL(2, 7), M12 , A6 : 2, A7 , PSL(2, 25), 25 :• S4, 24 : S3, PSL(2, 271), 12 x PSL(2, 13), and U(3, 7) : 2. We will present our construction of several of these images, including the Mathieu sporadic simple group M12 over the maximal subgroup PSL(2, 11), PSL(2, 17) over D9, PSL(2, 16) : 2 over [24 : 5] and PGL(2, 7) over S3. We will also give our method of finding isomorphism classes of images.

The goal of this presentation is to find original symmetric presentations of finite groups. It is frequently the case, that progenitors factored by appropriate relations produce simple and even sporadic groups as homomorphic images. We have discovered two of the twenty-six sporadic simple groups namely, M12, J1 and the Lie type group Suz(8). In addition several linear and classical groups will also be presented. We will present several progenitors including: 2*12: 22 x (3 : 2), 2*11: PSL2(11), 2*5: (5 : 4) which have produced the homomorphic images: M12 : 2, Suz(8) x 2, and J1 x 2. We will give monomial progenitors whose homomorphic images are: 17*10 :m PGL2(9), 3*4:m Z2 ≀D4 , and 13*2:m (22 x 3) : 2 which produce the homomorphic images:132 : ((2 x 13) : (2 x 3)), 2 x S9, and (22)•PGL4(3). Once we have a presentation of a group we can verify the group's existence through double coset enumeration. We will give proofs of isomorphism types of the presented images: S3 x PGL2(7) x S5, 28:A5, and 2•U4(2):2.

In 1990, Fröberg presented a combinatorial classification of the quadratic square-free monomial ideals with linear resolutions. He showed that the edge ideal of a graph has a linear resolution if and only if the complement of the graph is chordal. Since then, a generalization of Fröberg's theorem to higher dimensions has been sought in order to classify all square-free monomial ideals with linear resolutions. Such a characterization would also give a description of all square-free monomial ideals which are Cohen-Macaulay. In this thesis we explore one method of extending Fröberg's result. We generalize the idea of a chordal graph to simplicial complexes and use simplicial homology as a bridge between this combinatorial notion and the algebraic concept of a linear resolution. We are able to give a generalization of one direction of Fröberg's theorem and, in investigating the converse direction, find a necessary and sufficient combinatorial condition for a square-free monomial ideal to have a linear resolution over fields of characteristic 2.

Polynomial functions over finite fields are a major tool in computer science and electrical engineering and have a long history. Some of its aspects, like interpolation and permutation polynomials are described in this thesis. A complete characterization of subfield compatible polynomials (f in E[x] such that f(K) is a subset of L, where K,L are subfields of E) was recently given by J. Hull. In his work, he introduced the Frobenius permutation which played an important role. In this thesis, we fully describe the cycle structure of the Frobenius permutation. We generalize it to a permutation called a monomial permutation and describe its cycle factorization. We also derive some important congruences from number theory as corollaries to our work.

The aim of this thesis is to study the Hilbert function of a one-dimensional Gorenstein local ring of embedding dimension four in the case of monomial curves. We show that the Hilbert function is non-decreasing for some families of Gorenstein monomial curves in affine 4-space. In order to prove this result, under some arithmetic assumptions on generators of the defining ideal, we determine the minimal generators of their tangent cones by using the standard basis and check the Cohen-Macaulayness of them. Later, we determine the behavior of the Hilbert function of these curves, and we extend these families to higher dimensions by using a method developed by Morales. In this way, we obtain large families of local rings with non-decreasing Hilbert function.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-16T13:43:54Z No. of bitstreams: 1 arquivototal.pdf: 497004 bytes, checksum: f533fe667e534433904bf0bb58473fac (MD5) / Made available in DSpace on 2017-08-16T13:43:54Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 497004 bytes, checksum: f533fe667e534433904bf0bb58473fac (MD5) Previous issue date: 2015-08-28 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / Let k be an algebraically closed eld of characteristic zero and consider the polynomial ring S = k[x1, . . . , xn] endowed with the standard grading. The Waring's problem for a form F ∈ S of degree d asks about the least integer s ≥ 1 for which there exist linear forms L1, . . . , Ls ∈ S satisfying F = Σs i=1 Ldi. Such integer is called Waring rank of F. In this dissertation, we present a solution to this problem { due to Carlini-Catalisano-Geramita { in the case of monomials, as an interesting application of a theorem (due to the same authors) that establishes a lower bound for the multiplicity of (standard) graded, nitely generated, reduced, 1-dimensional k-algebras. / Seja k um corpo algebricamente fechado de caracter stica zero e considere o anel de polin^omios S = k[x1, . . . , xn] munido da gradua c~ao padr~ao. O Problema de Waring para uma forma F ∈ S de grau d questiona a respeito do menor inteiro s ≥ 1 para o qual existem formas lineares L1, . . . , Ls ∈ S satisfazendo F = Σs i=1 Ldi. Tal inteiro e denominado posto de Waring de F. Nesta disserta c~ao, apresentamos uma solu c~ao deste problema { devida a Carlini-Catalisano-Geramita { no caso de mon^omios, como uma interessante aplica c~ao de um teorema (dos mesmos autores) que estabelece uma cota inferior para a multiplicidade de k- algebras graduadas (padr~ao) nitamente geradas, reduzidas e 1-dimensionais.