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Computational algorithms for algebrasPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2009 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, Stockholm University , 2009. , 9 p.
##### Keyword [en]

Hilbert function, Gröbner basis, zero-dimensional ideal, affine variety, projective variety, run-time complexity
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-31552ISBN: 978-91-7155-974-6 (print)OAI: oai:DiVA.org:su-31552DiVA: diva2:277600
##### Public defence

2009-12-18, Sal 14, hus 5, Kräftriket, Stockholm, 10:00 (English)
##### Opponent

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##### Supervisors

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#####

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##### Note

At the time of doctoral defence, the following papers were unpublished and had a status as follows: Paper 3: Manuscript.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: ManuscriptAvailable from: 2009-11-26 Created: 2009-11-19 Last updated: 2009-11-20Bibliographically approved
##### List of papers

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.

1. Vector space bases associated to vanishing ideals of points$(function(){PrimeFaces.cw("OverlayPanel","overlay277499",{id:"formSmash:j_idt498:0:j_idt502",widgetVar:"overlay277499",target:"formSmash:j_idt498:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Complexity of comparing monomials and two improvements of the Buchberger-Möller algorithm$(function(){PrimeFaces.cw("OverlayPanel","overlay277498",{id:"formSmash:j_idt498:1:j_idt502",widgetVar:"overlay277498",target:"formSmash:j_idt498:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. A Buchberger like algorithm without monomial orderings: the graded commutative case$(function(){PrimeFaces.cw("OverlayPanel","overlay277497",{id:"formSmash:j_idt498:2:j_idt502",widgetVar:"overlay277497",target:"formSmash:j_idt498:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. An algorithm to determine the Hilbert series for graded associative algebras$(function(){PrimeFaces.cw("OverlayPanel","overlay277496",{id:"formSmash:j_idt498:3:j_idt502",widgetVar:"overlay277496",target:"formSmash:j_idt498:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Non-vanishing forms in projective space over finite fields$(function(){PrimeFaces.cw("OverlayPanel","overlay277495",{id:"formSmash:j_idt498:4:j_idt502",widgetVar:"overlay277495",target:"formSmash:j_idt498:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Multiplication matrices and ideals of projective dimension zero$(function(){PrimeFaces.cw("OverlayPanel","overlay277494",{id:"formSmash:j_idt498:5:j_idt502",widgetVar:"overlay277494",target:"formSmash:j_idt498:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1219",{id:"formSmash:lower:j_idt1219",widgetVar:"widget_formSmash_lower_j_idt1219",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1220_j_idt1222",{id:"formSmash:lower:j_idt1220:j_idt1222",widgetVar:"widget_formSmash_lower_j_idt1220_j_idt1222",target:"formSmash:lower:j_idt1220:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});