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.

We define the phylogenetic model of a trivalent graph as a generalization of a binary symmetric model of a trivalent phylogenetic tree. If the underlining graph is a tree, the model has a parametrization that can be expressed in terms of the tree. The model is always a polarized projective toric variety. Equivalently, it is a projective spectrum of a semigroup ring. We describe explicitly the generators of this projective coordinate ring for graphs with at most one cycle. We prove that models of graphs with the same topological invariants are deformation equivalent and share the same Hilbert function. We also provide an algorithm to compute the Hilbert function, which uses the structure of the graph as a sum of elementary ones. Also, this Hilbert function of phylogenetic model of a graph with g cycles is meaningful for the theory of connections on a Riemann surface of genus g.

This thesis is divided into two chapters. First, we want to study particularclasses of power ideals, with particular attention to their relation with the Fröberg conjecture on the Hilbert series of generic ideals. In the second part,we study a generalization (introduced by Fröberg, Ottaviani, and Shapiro in 2012)of the classical Waring problem for polynomials about writing homogeneouspolynomials as sums of powers. We see also how the theories of fat points andsecant varieties of Veronese varieties play a crucial role in the relation betweenthose chapters and in providing tools to nd an answer to our questions. The main results are the computation of the Hilbert series of particularclasses of power ideals, which in particular give us a proof of the Fröberg conjecturefor generic ideals generated by eight homogeneous polynomials of thesame degree in four variables, and the solution of the generalized Waring problemin the case of sums of squares in three and four variables. We also beginthe study of the generalized Waring problem for monomials.

We introduce localization and sheaves to define projective schemes, and in particular the projective n-space. Afterwards, we define closed subschemes of projective space and show that they arise from quotients of graded rings by homogeneous ideals. We then define the Hilbert function and Hilbert polynomial to determine several invariants of closed subschemes of projective space: their degree, dimension, and arithmetic genus. Finally, we provide numerous examples with explicit computations, finding the invariants of hypersurfaces, curves, the twisted cubic and more.

The goal of this thesis is to make Fröberg's conjecture more accessible to the average math graduate student by building up the necessary background material to understand specific examples where Fröberg's conjecture is true.

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.

A reproducing kernel Hilbert space (RKHS) is a Hilbert space $\mathscr{H}$ of functions with the property that the values $f(x)$ for $f \in \mathscr{H}$ are reproduced from the inner product in $\mathscr{H}$. Recent applications are found in stochastic processes (Ito Calculus), harmonic analysis, complex analysis, learning theory, and machine learning algorithms. This research began with the study of RKHSs to areas such as learning theory, sampling theory, and harmonic analysis. From the Moore-Aronszajn theorem, we have an explicit correspondence between reproducing kernel Hilbert spaces (RKHS) and reproducing kernel functions—also called positive definite kernels or positive definite functions. The focus here is on the duality between positive definite functions and their boundary spaces; these boundary spaces often lead to the study of Gaussian processes or Brownian motion. It is known that every reproducing kernel Hilbert space has an associated generalized boundary probability space. The Arveson (reproducing) kernel is $K(z,w) = \frac{1}{1-_{\C^d}}, z,w \in \B_d$, and Arveson showed, \cite{Arveson}, that the Arveson kernel does not follow the boundary analysis we were finding in other RKHS. Thus, we were led to define a new reproducing kernel on the unit ball in complex $n$-space, and naturally this lead to the study of a new reproducing kernel Hilbert space. This reproducing kernel Hilbert space stems from boundary analysis of the Arveson kernel. The construction of the new RKHS resolves the problem we faced while researching “natural” boundary spaces (for the Drury-Arveson RKHS) that yield boundary factorizations: \[K(z,w) = \int_{\mathcal{B}} K^{\mathcal{B}}_z(b)\overline{K^{\mathcal{B}}_w(b)}d\mu(b), \;\;\; z,w \in \B_d \text{ and } b \in \mathcal{B} \tag*{\it{(Factorization of} $K$).}\] Results from classical harmonic analysis on the disk (the Hardy space) are generalized and extended to the new RKHS. Particularly, our main theorem proves that, relaxing the criteria to the contractive property, we can do the generalization that Arveson's paper showed (criteria being an isometry) is not possible.

This thesis consists of six papers. In Paper I, we give an algorithm for merging sorted lists of monomials and together with a projection technique, we obtain a new complexity bound for the Buchberger-Möller algorithm and the FGLM algorithm. In Paper II, we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give complexity bounds. As an application we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks. In Paper III, we introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero. In Paper IV, we consider a subset of projective space over a finite field and give a geometric description of the minimal degree of a non-vanishing form with respect to this subset. We also give bounds on the minimal degree in terms of the cardinality of the subset. In Paper V, we study an associative version of an algorithm constructed to compute the Hilbert series for graded Lie algebras. In the commutative case we use Gotzmann's persistence theorem to show that the algorithm terminates in finite time. In Paper VI, we connect the commutative version of the algorithm in Paper V with the Buchberger algorithm. / At the time of doctoral defence, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / David Mumford introduced the concept of regularity of a coherent beam into the projective space in terms of local cohomology, generalizing a classic argument of Castelnuovo. In this dissertation under view of commutative algebra, we will introduce the concept of regularity of finitely generated graduated modules on the ring of polynomials. First, we perform a preliminary study on dimension theory and especially on Hilbert’s function. We also studied the basics of Cohen- Macaulay modules, properties of Betti’s graduated numbers, and the local cohomology functors. In the main chapter, we define the regularity of Castelnuovo-Mumford using the free resolution shifts. Soon after, we show that the definition of regularity can be given in terms of local cohomology, with emphasis on the cases of Artinian and Cohen-Macaulay modules. / David Mumford introduziu o conceito de regularidade de um feixe coerente no espac¸o projetivo em termos de cohomologia local, generalizando um argumento cl´assico de Castelnuovo. Nessa dissertac¸ ˜ao sob a vis˜ao da ´algebra comutativa, introduziremos o conceito de regularidade de m´odulos graduados finitamente gerados sobre o anel de polinˆomios. Primeiramente realizamos um estudo preliminar sobre teoria da dimens˜ao e em especial sobre a func¸ ˜ao de Hilbert. Tamb´em estudamos noc¸ ˜oes b´asicas em m´odulos Cohen-Macaulay, propriedades dos n´umeros graduados de Betti e dos funtores de cohomologia local. No cap´ıtulo principal, definimos a regularidade de Castelnuovo-Mumford utilizando os shifts de resoluc¸ ˜oes livres. Logo ap´os, mostramos que a definic¸ ˜ao de regularidade pode ser dada em termos de cohomologia local, dando ˆenfase aos casos de m´odulos Artinianos e Cohen-Macaulay. / São Cristóvão, SE

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This dissertation aims at a detailed study of the Hilbert function and graded Betti number and the statements of some theorems that relate these two theories. We will also a brief overview on free resolutions and minimal simplicial complex to demonstrate the theorem of Bayer, Sturmfels and Peeva and then, we will conclude with the following result: given an ideal J we will display a set X P2 such that the minimal resolution the ideal of de nition of X has the same Betti diagram of the minimal resolution of J. / Esta disserta¸c˜ao tem como objetivo um estudo detalhado da fun¸c˜ao de Hilbert e do n´umero graduado de Betti e as demonstra¸c˜oes de alguns teoremas que relacionam essas duas teorias. Faremos tamb´em um breve apanhado sobre resolu¸c˜oes livres minimais e complexo simplicial para demonstrar o teorema de Bayer, Peeva e Sturmfels e por fim e n˜ao menos importante concluiremos com o seguinte resultado: dado um ideal J exibiremos um conjunto X P2 tal que a resolu¸c˜ao minimal do ideal de defini¸c˜ao de X tenha o mesmo diagrama de Betti da resolu¸c˜ao minimal de J.