A note on Fröberg's conjecture for forms of equal degrees

Gleb Nenashev

doi:10.1016/j.crma.2017.01.011

Algebra/Algebraic geometry

A note on Fröberg's conjecture for forms of equal degrees
[Une note sur la conjecture de Fröberg pour des formes de degrés égaux]

Présenté par : the Editorial Board

Gleb Nenashev ¹

¹ Department of Mathematics, Stockholm University, 10691, Stockholm, Sweden

Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 272-276.

Résumé
Abstract

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.

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.

Reçu le : 2016-09-26
Accepté le : 2017-01-23
Publié le : 2017-02-07

DOI : 10.1016/j.crma.2017.01.011

Affiliations des auteurs :

Gleb Nenashev ¹

¹ Department of Mathematics, Stockholm University, 10691, Stockholm, Sweden

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Gleb Nenashev. A note on Fröberg's conjecture for forms of equal degrees. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 272-276. doi : 10.1016/j.crma.2017.01.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.011/

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