Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2011. / Submitted by wiliam de oliveira aguiar (wiliam@bce.unb.br) on 2011-06-27T17:20:02Z No. of bitstreams: 1 2011_TertulianoCarneirodeSouzaNeto.pdf: 489280 bytes, checksum: c757fc5257dd8408cbf6a1d641c6cbee (MD5) / Approved for entry into archive by Repositorio Gerência(repositorio@bce.unb.br) on 2011-06-30T17:50:48Z (GMT) No. of bitstreams: 1 2011_TertulianoCarneirodeSouzaNeto.pdf: 489280 bytes, checksum: c757fc5257dd8408cbf6a1d641c6cbee (MD5) / Made available in DSpace on 2011-06-30T17:50:48Z (GMT). No. of bitstreams: 1 2011_TertulianoCarneirodeSouzaNeto.pdf: 489280 bytes, checksum: c757fc5257dd8408cbf6a1d641c6cbee (MD5) / Seja f(x1, ..., xn) = a1xk 1 + ... + anxk n g(x1, ..., xn) = b1xk 1 + ... + bnxk n (1) um par de formas aditivas de grau pΤ (p − 1). Estamos interessados em obter condições que garantam a existência de zeros p-ádicos para o par (1). Uma conhecida conjectura, devida a Emil Artin, afirma que a condição n > 2k2 é suficiente. Utilizando técnicas da Teoria Combinatória dos Números, provamos que a condição n > 2 p (p/ P – 1) k2 − 2k é suficiente se k = 2.3Τ ou 4.5Τ, e em qualquer caso se Τ≥ (p – 1)/ 2. _____________________________________________________________________________________ ABSTRACT / Let f(x1, ..., xn) = a1xk 1 + ... + anxk n g(x1, ..., xn) = b1xk 1 + ... + bnxk n (1) be a pair of additive forms of degree pΤ (p − 1). We are interested in finding conditions which guarantee the existence of p-adic zeros to the pair (2). A well-known conjecture due to Emil Artin states that the condition n > 2k2 is sufficient. By means of techniques of Combinatorial Number Theory, we prove that n > 2 p (p/ P – 1) k2 − 2k is sufficient if k = 2.3Τ ou 4.5Τ, and in any case if Τ≥ (p – 1)/ 2.

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013. / Submitted by Albânia Cézar de Melo (albania@bce.unb.br) on 2013-09-09T14:24:13Z No. of bitstreams: 1 2013_LucianaLimaVentura.pdf: 572550 bytes, checksum: 0ce7cf628a3d83b89a7518122378820d (MD5) / Approved for entry into archive by Guimaraes Jacqueline(jacqueline.guimaraes@bce.unb.br) on 2013-09-09T15:46:43Z (GMT) No. of bitstreams: 1 2013_LucianaLimaVentura.pdf: 572550 bytes, checksum: 0ce7cf628a3d83b89a7518122378820d (MD5) / Made available in DSpace on 2013-09-09T15:46:43Z (GMT). No. of bitstreams: 1 2013_LucianaLimaVentura.pdf: 572550 bytes, checksum: 0ce7cf628a3d83b89a7518122378820d (MD5) / Uma versão da Conjectura de Artin afirma que para um sistema homogêneo com duas equações diagonais de grau k, cujos coeficientes são inteiros, ter solução p-ádica não trivial é suficiente que o número de variáveis seja maior que 2 k2. Nesse trabalho, vamos mostrar que a conjectura é verdadeira quando o grau é 2T . 3 ou 3T . 2, para T≥ 2. ______________________________________________________________________________ ABSTRACT / One version of Artin's Conjecture states that for a homogeneous system with two diagonal equations of degree k, whose coe cients are integers, exists a nontrivial p-adic solution provided the number of variables is greater than 2 k2. In this paper, we show that the conjecture is true when the degree is 2T . 3 or 3T . 2, for T≥ 2.

by Ka-hon Yeung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Bibliography: leaves 155-158. / Chapter 1) --- INTRODUCTION --- p.1 / Chapter 2) --- BACKGROUND MATERIALS AND DEFINITIONS --- p.3 / Chapter §1. --- THE RIEMANN ZETA FUNCTION --- p.3 / Chapter §2. --- THE DEDEKIND ZETA FUNCTION --- p.9 / Chapter §3. --- THE DIRICHLET L-FUNCTION --- p.11 / Chapter §4. --- PLACES AND ABSOLUTE VALUES --- p.13 / Chapter §5. --- THE HECKE L-FUNCTION --- p.14 / Chapter §6. --- CLASS FIELD THEORY --- p.17 / Chapter §7. --- LINEAR REPRESENTATIONS OF FINITE GROUPS --- p.19 / Chapter §8. --- THE ARTIN L-FUNCTION --- p.22 / Chapter 3) --- WORKS OF VARIOUS PEOPLE IN THE EVALUATION OF L-FUNCTIONS --- p.28 / Chapter §1. --- CLASS NUMBER FORMULA --- p.28 / Chapter §2. --- WORKS OF SHINTANI --- p.35 / Chapter §3. --- WORKS OF STARK --- p.65 / Chapter 4) --- STARK'S CONJECTURE --- p.90 / Chapter § 1. --- WORKS OF STARK --- p.90 / Chapter §2. --- WORKS OF TATE --- p.102 / Chapter §3. --- WORKS OF SANDS --- p.132 / NOTE --- p.146 / APPENDIX --- p.153 / BIBLIOGRAPHY --- p.155