In this thesis, we study Hilbert functions of monomial ideals in the polynomial ring and the Kruskal-Katona ring. In particular, we classify Gotzmann edge ideals and, more generally, Gotzmann squarefree monomial ideals. In addition, we discuss Betti numbers of Gotzmann ideals and measure how far certain edge ideals are from Gotzmann. This thesis also contains a thorough account the combinatorial relationship between lex segments and Macaulay representations of their dimensions and codimensions.

<div>In this thesis, we study invariants of graded modules over polynomial rings. In particular, we find bounds on the Hilbert functions and graded Betti numbers of certain modules. This area of research has been widely studied, and we discuss several well-known theorems and conjectures related to these problems. Our main results extend some known theorems from the case of homogeneous ideals of polynomial rings R to that of graded R-modules. In Chapters 2 & 3, we discuss preliminary material needed for the following chapters. This includes monomial orders for modules, Hilbert functions, graded Betti numbers, and generic initial modules.</div><div> </div><div> In Chapter 4, we discuss x_n-stability of submodules M of free R-modules F, and use this stability to examine properties of lexsegment modules. Using these tools, we prove our first main result: a general hypersurface restriction theorem for modules. This theorem states that, when restricting to a general hypersurface of degree j, the Hilbert series of M is bounded above by that of M^{lex}+x_n^jF. In Chapter 5, we discuss Hilbert series of local cohomology modules. As a consequence of our general hypersurface restriction theorem, we give a bound on the Hilbert series of H^i_m(F/M). In particular, we show that the Hilbert series of local cohomology modules of a quotient of a free module does not decrease when the module is replaced by a quotient by the lexicographic module M^{lex}.</div><div> </div><div> The content of Chapter 6 is based on joint work with Gabriel Sosa. The main theorem is an extension of a result of Caviglia and Sbarra to polynomial rings with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal with the following property: the ideal in the polynomial ring generated by the piecewise lex ideal, the ideal of powers, and the lex ideal has the same Hilbert function and Betti numbers at least as large as those of the original ideal. This bound on the Betti numbers is sharp, and is a closer bound than what was previously known in this setting.</div>

This thesis is constituted of two articles, both related to Hilbert functions and h-vectors. In the first paper, we deal with h-vectorsof reduced zero-dimensional schemes in the projective plane, and, in particular, with the problem of finding the possible h-vectors for the union of two sets of points of given h-vectors. In the second paper, we generalize the Green’s Hyperplane Restriction Theorem to the case of modules over the polynomial ring. / <p>QC 20131114</p>

Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial.

En utilisant le concept des fonctions de partition , nous étudions le comportement asymptotique des nombres de Betti gradués des puissances d’idéaux homogènes dans un polynôme sur un corp.Pour un Z-graduer positif, notre résultat principal affirme que les nombres de Betti des puissances est codé par un nombre fini des polynômes. Plus précisément, Z^2 peut être divisé en un nombre fini des régions telles que, dans chacun d’eux, dimk Tor^{S}_{i} (I^t,k)μ est un quasi-polynôme en (μ,t). Ce affine, dans une situation graduée, le résultat de Kodiyalam sur nombres de Betti des puissances dans [33].La déclaration principale traite le cas des produits des puissances d’idéaux homogènes dans un algèbre Z^d -graduée , pour un graduer positif, dans le sens de [37] et il est généralise également pour les filtrations I -good.Dans la deuxième partie, en utilisant la version paramétrique de l’algorithme de Barvinok, nous donnons une formule fermée pour les fonctions de Hilbert non-standard d’anneaux de polynômes, en petites dimensions. / Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. For a positive Z-grading, our main result states that the Betti numbers of powers is encoded by finitely many polynomials. More precisely, Z^2 can be splitted into a finite number of regions such that, in each of them, dim_k Tor^{S}_{i} (I^t,k)μ is a quasi-polynomial in (μ,t). This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers in [33]. The main statement treats the case of a power products of homogeneous ideals in a Z^d -graded algebra, for a positive grading, in the sense of [37] and it is also generalizes to I -good filtrations . In the second part , using the parametric version of Barvinok’s algorithm, we give a closed formula for non-standard Hilbert functions of polynomial rings, in low dimensions.