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A note on Fröberg's conjecture for forms of equal degrees
Stockholm University, Faculty of Science, Department of Mathematics.
2017 (English)In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 355, no 3, p. 272-276Article in journal (Refereed) Published
Abstract [en]

In this note we study ideals generated by generic forms in polynomial rings over any algebraicly closed field of characteristic zero. We prove for many cases that the (d+k)-th graded component of an ideal generated by generic forms of degree d has the expected dimension (given by dimension count). And as a consequence of our result, we obtain that ideals generated by several generic forms of degrees d usually have the expected Hilbert series. The precise form of this expected Hilbert series, in general, is known as Fröberg's conjecture.

Abstract [fr]

Dans cette note, nous étudions les idéaux générés par des formes génériques dans des anneaux de polynômes sur un champ algébriquement clos de caractéristique nulle. Nous montrons que, dans de nombreux cas, la (d+k)-ième composante graduelle d'un idéal engendré par les formes génériques de degré d a la dimension attendue (donnée par certains calculs). Comme une conséquence de notre résultat, nous obtenons que les idéaux générés par plusieurs formes génériques de degré d ont habituellement la série de Hilbert prévue. Cette dernière affirmation est connue comme la conjecture de Fröberg.

Place, publisher, year, edition, pages
2017. Vol. 355, no 3, p. 272-276
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-139896DOI: 10.1016/j.crma.2017.01.011ISI: 000397097700006OAI: oai:DiVA.org:su-139896DiVA, id: diva2:1075398
Available from: 2017-02-19 Created: 2017-02-19 Last updated: 2018-04-07Bibliographically approved
In thesis
1. Around power ideals: From Fröberg's conjecture to zonotopal algebra
Open this publication in new window or tab >>Around power ideals: From Fröberg's conjecture to zonotopal algebra
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study power algebras, which are quotient of polynomial rings by power ideals. We will study Hilbert series of such ideals and their other properties. We consider two important special cases, namely, zonotopal ideals and generic ideals. Such ideals have a lot combinatorial properties.

In the first chapter we study zonotopal ideals, which were defined and used in several earlier publications. The most important works are by F.Ardila and A.Postnikov and by O.Holtz and A.Ron. These papers originate from different sources, the first source is homology theory, the second one is the theory of box splines. We study quotient algebras by these ideals; these algebras have a nice interpretation for their Hilbert series, as specializations of their Tutte polynomials. There are two important subclasses of these algebras, called unimodular and graphical. The graphical algebras were defined by A.Postnikov and B.Shapiro. In particular, the external algebra of a complete graph is exactly the algebra generated by the Bott-Chern forms of the corresponding complete flag variety. One of the main results of the thesis is a characterization of external algebras. In fact, for the case of graphical and unimodular algebras we prove that external algebras are in one-to-one correspondence with graphical and regular matroids, respectively.

In the second chapter we study Hilbert series of generic ideals. By a generic ideal we mean an ideal generated by forms from some class, whose coefficients belong to a Zariski-open set. There are two main classes to consider: the first class is when we fix the degrees of generators; the famous Fröberg's conjecture gives the expected Hilbert series of such ideals; the second class is when an ideal is generated by powers of generic linear forms. There are a few partial results on Fröberg's conjecture, namely, when the number of variables is at most three. In both classes the Hilbert series is known in the case when the number of generators is at most (n+1). In both cases we construct a lot of examples when the degree of generators are the same and the Hilbert series is the expected one.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2018. p. 58
National Category
Discrete Mathematics Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-154903 (URN)978-91-7797-244-0 (ISBN)978-91-7797-245-7 (ISBN)
Public defence
2018-05-25, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript. Paper 7: Manuscript. Paper 8: Manuscript.

Available from: 2018-05-02 Created: 2018-04-07 Last updated: 2018-05-07Bibliographically approved

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