| 1. |
- Lundqvist, Samuel, 1978-
(författare)
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Computational algorithms for algebras
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 2. |
- Oneto, Alessandro, 1988-
(författare)
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Power ideals, Fröberg conjecture and Waring problems
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 3. |
- Emtander, Eric, 1979-
(författare)
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Chordal and Complete Structures in Combinatorics and Commutative Algebra
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is divided into two parts. The first part is concerned with the commutative algebra of certain combinatorial structures arising from uniform hypergraphs. The main focus lies on two particular classes of hypergraphs called chordal hypergraphs and complete hypergraphs, respectively. Both these classes arise naturally as generalizations of the corresponding well known classes of simple graphs. The classes of chordal and complete hypergraphs are introduced and studied in Chapter 2 and Chapter 3 respectively. Chapter 4, that is the content of \cite{E5}, answers a question posed at the P.R.A.G.MAT.I.C. summer school held in Catania, Italy, in 2008. In Chapter 5 we study hypergraph analogues of line graphs and cycle graphs. Chapter 6 is concerned with a connectedness notion for hypergraphs and in Chapter 7 we study a weak version of shellability.The second part is concerned with affine monoids and their monoid rings. Chapter 8 provide a combinatorial study of a class of positive affine monoids that behaves in some sense like numerical monoids. Chapter 9 is devoted to the class of numerical monoids of maximal embedding dimension. A combinatorial description of the graded Betti numbers of the corresponding monoid rings in terms of the minimal generators of the monoids is provided. Chapter 10 is concerned with monomial subrings generated by edge sets of complete hypergraphs.
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| 4. |
- Nonkane, Ibrahim, 1982-
(författare)
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Decomposition of modules over the Weyl algebra
- 2013
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The thesis consists of two papers that treat decomposition of modules over the Weyl algebra. In the first paper decomposition of holonomic modules is used to prove that there is a finite set of Noetherian operators that suffice to distinguish the elements in a primary ideal. In the second the decomposition of the direct image of the root-to-coefficients mapping are constructed using the representation theory of the symmetric group, in particular higher Specht polynomials.
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| 5. |
- Emtander, Eric, 1979-
(författare)
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On hypergraph algebras
- 2008
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)
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