In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the existence and properties of the model companion of the theory of partial differential fields with an automorphism. The approach taken here to these subjects is to relativize the algebro geometric notions of prolongation and D-variety to differential notions with respect to a fixed differential structure. It is shown that every differential algebraic group which is not of maximal differential type is definably isomorphic to the sharp points of a relative D-group. Pillay's generalized finite dimensional differential Galois theory is extended to the possibly infinite dimensional partial setting. Logarithmic differential equations on relative D-groups are discussed and the associated differential Galois theory is developed. The notion of generalized strongly normal extension is naturally extended to the partial setting, and a connection betwen these extensions and the Galois extensions associated to logarithmic differential equations is established. A geometric characterization, in the spirit of Pierce-Pillay, for the theory DCF_{0,m+1} (differentially closed fields of characteristic zero with m+1 commuting derivations) is given in terms of the differential algebraic geometry of DCF_{0,m} using relative prolongations. It is shown that this characterization can be rephrased in terms of characteristic sets of prime differential ideals, yielding a first-order geometric axiomatization of DCF_{0,m+1}. Using the machinery of characteristic sets of prime differential ideals it is shown that the theory of partial differential fields with an automorphism has a model companion. Some basic model theoretic properties of this theory are presented: description of its completions, supersimplicity and elimination of imaginaries. Differential-difference modules are introduced and they are used, together with jet spaces, to establish the canonical base property for finite dimensional types, and consequently the Zilber dichotomy for minimal finite dimensional types.

Given a matrix A in F(t)[D;\delta]^{n\times n} over the ring of differential polynomials, we first prove the existence of the Hermite form H of A over this ring. Then we determine degree bounds on U and H such that UA=H. Finally, based on the degree bounds on U and H, we compute the Hermite form H of A by reducing the problem to solving a linear system of equations over F(t). The algorithm requires a polynomial number of operations in F in terms of the input sizes: n, deg_{D} A, and deg_{t} A. When F=Q it requires time polynomial in the bit-length of the rational coefficients as well.

Given a matrix A in F(t)[D;\delta]^{n\times n} over the ring of differential polynomials, we first prove the existence of the Hermite form H of A over this ring. Then we determine degree bounds on U and H such that UA=H. Finally, based on the degree bounds on U and H, we compute the Hermite form H of A by reducing the problem to solving a linear system of equations over F(t). The algorithm requires a polynomial number of operations in F in terms of the input sizes: n, deg_{D} A, and deg_{t} A. When F=Q it requires time polynomial in the bit-length of the rational coefficients as well.

In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the existence and properties of the model companion of the theory of partial differential fields with an automorphism. The approach taken here to these subjects is to relativize the algebro geometric notions of prolongation and D-variety to differential notions with respect to a fixed differential structure. It is shown that every differential algebraic group which is not of maximal differential type is definably isomorphic to the sharp points of a relative D-group. Pillay's generalized finite dimensional differential Galois theory is extended to the possibly infinite dimensional partial setting. Logarithmic differential equations on relative D-groups are discussed and the associated differential Galois theory is developed. The notion of generalized strongly normal extension is naturally extended to the partial setting, and a connection betwen these extensions and the Galois extensions associated to logarithmic differential equations is established. A geometric characterization, in the spirit of Pierce-Pillay, for the theory DCF_{0,m+1} (differentially closed fields of characteristic zero with m+1 commuting derivations) is given in terms of the differential algebraic geometry of DCF_{0,m} using relative prolongations. It is shown that this characterization can be rephrased in terms of characteristic sets of prime differential ideals, yielding a first-order geometric axiomatization of DCF_{0,m+1}. Using the machinery of characteristic sets of prime differential ideals it is shown that the theory of partial differential fields with an automorphism has a model companion. Some basic model theoretic properties of this theory are presented: description of its completions, supersimplicity and elimination of imaginaries. Differential-difference modules are introduced and they are used, together with jet spaces, to establish the canonical base property for finite dimensional types, and consequently the Zilber dichotomy for minimal finite dimensional types.

Orientador: Paulo Roberto Brumatti / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T19:30:30Z (GMT). No. of bitstreams: 1 Veloso_MarceloOliveira_D.pdf: 662198 bytes, checksum: f119f8026ebe09649fca4a175b7cec47 (MD5) Previous issue date: 2009 / Resumo: Este trabalho é dedicado ao estudo das derivações localmente nilpotentes de certas K-álgebras finitamente geradas, onde K é um corpo de característica zero. Estes domínios são generalizações de anéis bem conhecidos na literatura sendo um deles o anel de Fermat. Mais precisamente, caracterizamos o conjunto das derivações localmente nilpotentes destes domínios ou de um subconjunto deste conjunto. Também calculamos o ML invariante destes domínios e como aplicação direta destas informações encontramos um conjunto de geradores para o grupo dos automorfismos de um destes domínos. No caso do anel de Fermat mostramos que nem sempre temos um domíno rígido. Além disso, verificamos que a Conjectura de Nakai é verdadeira para o anel de Fermat. / Abstract: This work is dedicated to the study of locally nilpotent derivations of certain finitely generated K-algebras, where K is a field of zero characteristic. These domains are generalizations of the well-known rings in the literature. One of this is the Fermat ring. More precisely, we characterize the set of locally nilpotent derivations of these domains or some subsets of this set. We also calculate the ML invariant of these domains and as a direct application of these results we find a set of generators for the group of automorphisms of some of these domains. We show that the Fermat ring is not always a rigid domain. Furthermore, we prove that Nakai's conjecture is true for the ring Fermat. / Doutorado / Algebra Comutativa / Doutor em Matemática

Depuis les années 90, la méthode de création télescopique de Zeilberger a joué un rôle important dans la preuve automatique d'identités mettant en jeu des fonctions spéciales. L'objectif de long terme que nous attaquons dans ce travail est l'obtension d'algorithmes et d'implantations rapides pour l'intégration et la sommation définies dans le cadre de cette création télescopique. Nos contributions incluent de nouveaux algorithmes pratiques et des critères théoriques pour tester la terminaison d'algorithmes existants. Sur le plan pratique, nous nous focalisons sur la construction de télescopeurs minimaux pour les fonctions rationnelles en deux variables, laquelle a de nombreuses applications en lien avec les fonctions algébriques et les diagonales de séries génératrices rationnelles. En considérant cette classe d'entrées contraintes, nous parvenons à mâtiner la méthode générale de création télescopique avec réduction bien connue d'Hermite, issue de l'intégration symbolique. En outre, nous avons obtenu pour cette sous-classe quelques améliorations des algorithmes classiques d'Almkvist et Zeilberger. Nos résultats expérimentaux ont montré que les algorithmes à base de réduction d'Hermite battent tous les autres algorithmes connus, à la fois en ce qui concerne la complexité au pire et en ce qui concerne les mesures de temps sur nos implantations. Sur le plan théorique, notre premier résultat est motivé par la conjecture de Wilf et Zeilberger au sujet des fonctions hyperexponentielles-hypergéométriques holonomes. Nous présentons un théorème de structure pour les fonctions hyperexponentielles-hypergéométriques de plusieurs variables, indiquant qu'une telle fonction peut s'écrire comme le produit de fonctions usuelles. Ce théorème étend à la fois le théorème d'Ore et Sato pour les termes hypergéométriques en plusieurs variables et le résultat récent par Feng, Singer et Wu. Notre second résultat est relié au problème de l'existence de télescopeurs. Dans le cas discret à deux variables, Abramov a obtenu un critère qui indique quand un terme hypergéométrique a un télescopeur. Des résultats similaires ont été obtenus pour le $q$-décalage par Chen, Hou et Mu. Ces résultats sont fondamentaux pour la terminaison des algorithmes s'inspirant de celui de Zeilberger. Dans les autres cas mixtes continus/discrets, nous avons obtenu deux critères pour l'existence de télescopeurs pour des fonctions hyperexponentielles-hypergéométriques en deux variables. Nos critères s'appuient sur une représentation standard des fonctions hyperexponentielles-hypergéométriques en deux variables, sur sur deux décompositions additives.

Orientador: Paulo Roberto Brumatti / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica / Made available in DSpace on 2018-08-10T20:04:54Z (GMT). No. of bitstreams: 1 Bianchi_AngeloCalil_M.pdf: 609753 bytes, checksum: d05b2d15e03b1e36b83018ed28d8da63 (MD5) Previous issue date: 2008 / Resumo: O objetivo geral desse trabalho é entender a teoria das bases SAGBI num ponto de vista estrutural, buscando critérios para sua existência e resultados que comprovem sua eficácia para o estudo de certas k-subalgebras de k[x], bem como estudar a teoria geral das derivações sobre anéis de polinômios, suas localizações e quocientes, visando explorar as propriedades algébricas do núcleo destas derivações e as estruturas das k-subalgebras de k[x] que podem ser vistas como tais núcleos. O objetivo específico é estudar a teoria algébrico-geométrica para k-derivações em k[x], desenvolvida por Shigeru Kuroda, e utilizar dessa teoria para estabelecer uma condição para que o núcleo de uma tal derivação seja uma k-subalgebra finitamente gerada e outra para que este possua uma base SAGBI finita. Em cada momento ao longo do trabalho também é desejado enfatizar o comportamento das k-derivações que são localmente nilpotentes e obter uma forma algorítmica para determinar os geradores de seus núcleos, no caso particular da derivação ao possuir uma slice / Abstract: The general objective of this work is to understand the SAGBI bases theory from a structural point of view, seeking criterias for it¿s existence and results that prove it¿s effitiency in the study of certain subalgebras of k[x], as well as to study the general theory of derivations over polynomial rings, it¿s localizations and quotients, in order to explore the algebraic properties of the kernel of this derivations and the structures of the k-subalgebras of k[x] that may be seen as such kernels. The specific objective is to study the algebraic-geometric theory of k-derivations in k[x], developed by Shigeru Kuroda, and to use this theory to stabilish a condition for the kernel of one such derivation to be a finitely generated k-subalgebra and another condition for this derivation to have finite SAGBI base. Along this work we also want to emphasize the behavior of locally nilpotent k-derivations and to obtain an algorithmic way to determine the generators of it¿s kernels, in the particular case that the derivation has a slice / Mestrado / Matematica / Mestre em Matemática

This thesis studies various aspects of differential algebra, from fundamental concepts to practical computations. A characteristic feature of this work is the use of combinatorial techniques, which offer a unique and original perspective on the subject matter. First, we establish the connection between the n-jet space of the fat point defined by xm and the stable set polytope of a perfect graph. We prove that the dimension of the coordinate ring of the scheme defined by polynomial arcs of degree less than or equal to n is a polynomial in m of degree n + 1. This is based on Zobnin’s result which states that the set {x^m} is a differential Gr ̈obner basis for its differential ideal. We generalize this statement to the case of two independent variables and link the dimensions in this case to some triangulations of the p × q rectangle, where the pair (p, q) now plays the role of n. Second, we study the arc space of the fat point x^m on a line from the point of view of filtration by finite-dimensional differential algebras. We prove that the generating series of the dimensions of these differential algebras is m/(1 -mt) . Based on this we propose a definition of the multiplicity of a solution of an algebraic differential equation as the growth of the dimensions of these differential algebras. This generalizes the concept of the multiplicity of an ideal in a polynomial ring. Furthermore, we determine a full description of the set of standard monomials of the differential ideal generated by x^m. This description proves a conjecture by Afsharijoo concerning a new version of the Roger-Ramanujan identities. Every homogeneous linear system of partial differential equations with constant coef- ficients can be encoded by a submodule of the ring of polynomials. We develop practical methods for computing the space of solutions to these PDEs. These spaces are typically infinite dimensional, and we use the Ehrenpreis–Palamodov Theorem for finite encoding. We apply this finite encoding to the solutions of the PDEs associated with the arc spaces of a double point. We prove that these vector spaces are spanned by determinants of some special Wronskians, and we relate them to differentially homogeneous polynomials. Finally, we introduce D-algebraic functions: they are solutions to algebraic differential equations. We study closure properties of these functions. We present practical algorithms and their implementations for carrying out arithmetic operations on D-algebraic functions. This amounts to solving elimination problems for differential ideals.

In this thesis we study Ax-Schanuel type inequalities for abstract differential equations. A motivating example is the exponential differential equation. The Ax-Schanuel theorem states positivity of a predimension defined on its solutions. The notion of a predimension was introduced by Hrushovski in his work from the 1990s where he uses an amalgamation-with-predimension technique to refute Zilber's Trichotomy Conjecture. In the differential setting one can carry out a similar construction with the predimension given by Ax-Schanuel. In this way one constructs a limit structure whose theory turns out to be precisely the first-order theory of the exponential differential equation (this analysis is due to Kirby (for semiabelian varieties) and Crampin, and it is based on Zilber's work on pseudo-exponentiation). One says in this case that the inequality is adequate. Thus, by an Ax-Schanuel type inequality we mean a predimension inequality for a differential equation. Our main question is to understand for which differential equations one can find an adequate predimension inequality. We show that this can be done for linear differential equations with constant coefficients by generalising the Ax-Schanuel theorem. Further, the question turns out to be closely related to the problem of recovering the differential structure in reducts of differentially closed fields where we keep the field structure (which is quite an interesting problem in its own right). So we explore that question and establish some criteria for recovering the derivation of the field. We also show (under some assumptions) that when the derivation is definable in a reduct then the latter cannot satisfy a non-trivial adequate predimension inequality. Another example of a predimension inequality is the analogue of Ax-Schanuel for the differential equation of the modular j-function due to Pila and Tsimerman. We carry out a Hrushovski construction with that predimension and give an axiomatisation of the first-order theory of the strong Fraïssé limit. It will be the theory of the differential equation of j under the assumption of adequacy of the predimension. We also show that if a similar predimension inequality (not necessarily adequate) is known for a differential equation then the fibres of the latter have interesting model theoretic properties such as strong minimality and geometric triviality. This, in particular, gives a new proof for a theorem of Freitag and Scanlon stating that the differential equation of j defines a trivial strongly minimal set.

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