The negative answer to Kameko's conjecture on the hit problem

Nguyễn Sum

doi:10.1016/j.crma.2010.03.021

Topology

The negative answer to Kameko's conjecture on the hit problem
[Un contre-exemple à la conjecture de Kameko]

Présenté par : Christophe Soulé

Nguyễn Sum ¹

¹ Department of Mathematics, University of Quynhon, 170 An Duong Vuong, Quynhon, Vietnam

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 669-672.

Résumé
Abstract

Cette Note donne un contre-exemple à la conjecture de Kameko. Celle ci donnait une borne supérieure explicite pour le cardinal d'un système minimal de générateurs – comme module sur l'algèbre de Steenrod $A$ – de l'algèbre polynomiale $P_{k}$ en k générateurs (de degré 1) sur le corps $F_{2}$ . La conjecture est vraie pour $k = 3$ (Kameko, thèse Johns Hopkins University, 1990), récemment démontrée par Kameko, Nam et l'auteur de la Note pour $k = 4$ , qui montre ici qu'elle est fausse pour $k > 4$ . Pour donner ce contre-exemple l'auteur se restreint à certains degrés, et démontre une formule de récurrence pour le cardinal d'un système minimal de générateurs en ces degrés. C'est alors une conséquence facile de cette formule qui montre que la conjecture est fausse si $k > 4$ .

This Note gives a counter-example to Kameko's conjecture, stating an explicit upper bound for the cardinal of a minimal system of generators – as module over the Steenrod's algebra $A$ – of the polynomial algebra $P_{k}$ in k generators (of degree 1) over the field $F_{2}$ . The conjecture is true for $k = 3$ (Kameko (1990) [6]), $k = 4$ (Kameko (2003) [7] and the author of this Note; Sum (preprint) [15]), but false for $k > 4$ . In order to give the counter-example we restrict to some degrees and prove a recurrence relation for the cardinal of a minimal system of generators in these degrees. It results as an easy consequence that the conjecture is false for $k > 4$ .

Reçu le : 2009-04-13
Accepté le : 2009-11-23
Publié le : 2010-05-06

DOI : 10.1016/j.crma.2010.03.021

Affiliations des auteurs :

Nguyễn Sum ¹

¹ Department of Mathematics, University of Quynhon, 170 An Duong Vuong, Quynhon, Vietnam

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Nguyễn Sum. The negative answer to Kameko's conjecture on the hit problem. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 669-672. doi : 10.1016/j.crma.2010.03.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.021/

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