Definitividade de formas quadráticas – uma abordagem polinomial

• Ano de defesa: 2016
• Autor(a) principal: Alves, Jesmmer da Silveira
• Orientador(a): Castonguay, Diane
• Banca de defesa: Castonguay, Diane , Centeno, Carmen, Alvares, Edson Ribeiro, Martinez, Fabio Henrique Viduani, Longo, Humberto José
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Goiás
• Programa de Pós-Graduação: Programa de Pós-graduação em Ciência da Computação (INF)
• Departamento: Instituto de Informática - INF (RG)
• País: Brasil
• Palavras-chave em Português: Formas quadráticas; Redução de Gauss; Formas unitárias; Formas críticas; Formas hipercríticas; Algoritmo polinomial
• Palavras-chave em Inglês: Quadratic forms; Gauss reduction; Unit form; Critical restrictions; Hypercritical restrictions; Polynomial algorithm
• Área do conhecimento CNPq: CIENCIA DA COMPUTACAO::MATEMATICA DA COMPUTACAO
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/6586

Quadratic forms are algebraic expressions that have important role in different areas of computer science, mathematics, physics, statistics and others. We deal with rational quadratic forms and integral quadratic forms, with rational and integer coefficients respectively. Existing methods for recognition of rational quadratic forms have exponential time complexity or use approximation that weaken the result reliability. We develop a polinomial algorithm that improves the best-case of rational quadratic forms recognition in constant time. In addition, new strategies were used to guarantee the results reliability, by representing rational numbers as a fraction of integers, and to identify linear combinations that are linearly independent, using Gauss reduction. About the recognition of integral quadratic forms, we identified that the existing algorithms have exponential time complexity for weakly nonnegative type and are polynomial for weakly positive type, however the degree of the polynomial depends on the algebra dimension and can be very large. We have introduced a polynomial algorithm for the recognition of weakly nonnegative quadratic forms. The related algorithm identify hypercritical restrictions testing every subgraph of 9 vertices of the quadratic form associated graph. By adding Depth First Search approach, a similar strategy was used in the recognition of weakly positive type. We have also shown that the recognition of integral quadratic forms can be done by mutations in the related exchange matrix.