Generalising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and define the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building up a general theory of involutive bases of polynomial ideals. In particular, we give algorithms for checking the involutive basis property and for completing an arbitrary basis to an involutive one. It turns out that our theory is more constructive and more exible than the axiomatic approach to general involutive bases due to Gerdt and Blinkov. Finally, we show that an involutive basis contains more structural information about the ideal of leading terms than a Gröbner basis and that it is straight forward to compute the (affine) Hilbert function of an ideal I from an arbitrary involutive basis of I.

In [ASTW] it was given a Gröbner reduction based division formula for entire functions by polynomial ideals. Here we give degree bounds where the input function can be truncated in order to compute approximations of the coeffcients of the power series appearing in the division formula within a given precision. In addition, this method can be applied to the approximation of the value of the remainder function at some point.

In [ASTW] it was given a Gröbner reduction based division formula for entire functions by polynomial ideals. Here we give degree bounds where the input function can be truncated in order to compute approximations of the coeffcients of the power series appearing in the division formula within a given precision. In addition, this method can be applied to the approximation of the value of the remainder function at some point.

Commutative Gröbner bases theory is well known and widely used. In this thesis, we will discuss thoroughly its generalization to noncommutative polynomial ring <em>k</em><<em>X</em>> which is also an associative free algebra. We introduce some results on monomial orders due to John Lawrence and the author. We show that a noncommutative monomial order is a well order while a one-sided noncommutative monomial order may not be. Then we discuss the generalization of polynomial reductions, S-polynomials and the characterizations of noncommutative Gröbner bases. Some results due to Mora are also discussed, such as the generalized Buchberger's algorithm and the solvability of ideal membership problem for homogeneous ideals. At last, we introduce Newman's diamond lemma and Bergman's diamond lemma and show their relations with Gröbner bases theory.

Commutative Gröbner bases theory is well known and widely used. In this thesis, we will discuss thoroughly its generalization to noncommutative polynomial ring <em>k</em><<em>X</em>> which is also an associative free algebra. We introduce some results on monomial orders due to John Lawrence and the author. We show that a noncommutative monomial order is a well order while a one-sided noncommutative monomial order may not be. Then we discuss the generalization of polynomial reductions, S-polynomials and the characterizations of noncommutative Gröbner bases. Some results due to Mora are also discussed, such as the generalized Buchberger's algorithm and the solvability of ideal membership problem for homogeneous ideals. At last, we introduce Newman's diamond lemma and Bergman's diamond lemma and show their relations with Gröbner bases theory.

This paper covers basic theory of Grobner Bases and an algebraic analysis of the linear constant coefficient partial differential operators, specifically the Cauchy-Fueter operator. We will review examples and theory of regular and biregular functions in several quaternionic variables. / text

Made available in DSpace on 2018-08-01T22:30:14Z (GMT). No. of bitstreams: 1 tese_6990_Correção_dissert_Flavio_28_10 (1).pdf: 746942 bytes, checksum: d24587836883c9d3d4a30408eb30dc57 (MD5) Previous issue date: 2015-08-21 / O estudo de Equações Diferenciais sob o ponto de vista algébrico foi iniciado por volta de 1932 por J. F. Ritt com o trabalho "Differential Equations from de Algebraic Standpoint". Posteriormente, em 1948, Ritt publicou o livro "Differential Algebra". Trata-se de se utilizar as ferramentas da Geometria Algébrica e da Álgebra Comutativa para estudar operadores diferenciais e consequentemente, conjuntos soluções de sistemas de equações diferenciais parciais. Nesta dissertação, pretendemos fazer um estudo preliminar da Álgebra Diferencial com o objetivo principal de aplicar em problemas de soluções de equações diferenciais parciais. Os trabalhos de referências principais são Implicitization of differential rational parametric equations de X. S. Gao (2003) e Implicitization of DPPES and Deifferential Resultants de S. L. Rueda e J. R. Sendra (2008).

Estudaremos a teoria de bases de Gröbner em anéis de polinômios comutativos com coeficientes em uma nelnoetheriano e em anéis de operadores diferencias. Apresentaremos, em ambos casos, uma generalização do algoritmo da divisão, do S-polinômio e do algoritmo de Buchberger para calcular bases de Gröbner. / We study the theory of Gröbner bases for commutative polynomials rings over a noetherian ring and of rings of differential operators. In both cases we exhibit a generalization of the division algorithm, the S -polynomial and the Buchberger algorithm for computing Gröbner bases.

Estudaremos a teoria de bases de Gröbner em anéis de polinômios comutativos com coeficientes em uma nelnoetheriano e em anéis de operadores diferencias. Apresentaremos, em ambos casos, uma generalização do algoritmo da divisão, do S-polinômio e do algoritmo de Buchberger para calcular bases de Gröbner. / We study the theory of Gröbner bases for commutative polynomials rings over a noetherian ring and of rings of differential operators. In both cases we exhibit a generalization of the division algorithm, the S -polynomial and the Buchberger algorithm for computing Gröbner bases.

Les nombres p-adiques sont un analogue des nombres réels plus proche de l’arithmétique. L’avènement ces dernières décennies de la géométrie arithmétique a engendré la création de nombreux algorithmes utilisant ces nombres. Ces derniers ne peuvent être de manière générale manipulés qu’à précision finie. Nous proposons une méthode, dite de précision différentielle, pour étudier ces problèmes de précision. Elle permet de se ramener à un problème au premier ordre. Nous nous intéressons aussi à la question de savoir quelles bases de Gröbner peuvent être calculées sur les p-adiques. / P-Adic numbers are a field in arithmetic analoguous to the real numbers. The advent during the last few decades of arithmetic geometry has yielded many algorithms using those numbers. Such numbers can only by handled with finite precision. We design a method, that we call differential precision, to study the behaviour of the precision in a p-adic context. It reduces the study to a first-order problem. We also study the question of which Gröbner bases can be computed over a p-adic number field.