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.

The thesis is a collection of four papers dealing with Hilbert functions and Betti numbers.In the first paper, we study the h-vectors of reduced zero-dimensional schemes in . In particular we deal with the problem of findingthe possible h-vectors for the union of two sets of points of given h-vectors. To answer to this problem, we give two kinds of bounds on theh-vectors and we provide an algorithm that calculates many possibleh-vectors.In the second paper, we prove a generalization of Green’s Hyper-plane Restriction Theorem to the case of finitely generated modulesover the polynomial ring, providing an upper bound for the Hilbertfunction of the general linear restriction of a module M in a degree dby the corresponding Hilbert function of a lexicographic module.In the third paper, we study the minimal free resolution of theVeronese modules, , where by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that is Cohen-Macaulay if and only if k < d, and that its minimal resolutionis linear when k > d(n − 1) − n. We prove combinatorially that the resolution of is pure. We provide a formula for the Hilbert seriesof the Veronese modules. As an application, we calculate the completeBetti diagrams of the Veronese rings .In the fourth paper, we apply the same combinatorial techniques inthe study of the properties of pinched Veronese rings, in particular weshow when this ring is Cohen-Macaulay. We also study the canonicalmodule of the Veronese modules.

3.

Greco, Ornella

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this paper, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of the squarefree divisor complex. In particular, we study the Cohen-Macaulay property of these rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

4.

Greco, Ornella

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

In this paper, we prove a generalization of Green's Hyperplane RestrictionTheorem to the case of modules over the polynomial ring, providing in particularan upper bound for the Hilbert function of the general linear restrictionof a module M in a degree d by the corresponding Hilbert function of alexicographic module.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Martino, Ivan

Syzygies of the Veronese Modules2016In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, no 9, p. 3890-3906Article in journal (Refereed)

Abstract [en]

We study the minimal free resolution of the Veronese modules, S-n,S- d,S- k = circle plus S-i >= 0(k+id), where S = K[x(1), ... , x(n)], by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We prove that S-n,S- d,S- k is Cohen-Macaulay if and only if k < d, and that its minimal resolution is pure and has some linearity features when k > d (n - 1) - n. We prove combinatorially that the resolution of S-2,S- d,S- k is pure. We show that HS(S-n,S- d,S- k; z) = 1/(n-1)d(n-1)/dz(n-1) [z(k+n-1)/1-z(d)]. As an application, we calculate the complete Betti diagrams of the Veronese rings K[x, y, z]((d)), for d = 4, 5, and K[x, y, z, u]((3)).

Given two h-vectors, handh0, we study which are the possible h-vectors for the union of two disjoint sets of points in P2, respectively associated to h and h0and how they can be constructed. We will give some bounds for the resulting h-vector and we will show how to construct the minimal h-vector of the union among all possible ones.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Conca, P.

Cutello, V.

Pavone, M.

Nicosia, G.

Packing equal disks in a unit square: An immunological optimization approach2014In: International Workshop on Artificial Immune Systems, AIS 2015/ICSI3 2015 - Systems Immunology, Immunoinformatics and Immune-computation: Immunology without Borders, Proceedings, Institute of Electrical and Electronics Engineers (IEEE), 2014, article id 7469239Conference paper (Refereed)

Abstract [en]

Packing equal disks in a unit square is a classical geometrical problem which arises in many industrial and scientific fields. Finding optimal solutions has been proved to be NPhard, therefore, only local optimal solutions can be identified. We tackle this problem by means of the optimization Immune Algorithm (optIA), which has been proved to be among the best derivativefree optimization algorithms. In particular, OPTIA is used to pack up to 150 disks in a unit square. Experimental results show that the immune algorithm is able to locate the putative global optimum for all the instances. Moreover, a comparison with the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) shows that OPTIA is more robust.