On a conjecture of Kato and Kuzumaki

Izquierdo, Diego

doi:10.2140/ant.2018.12.429

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On a conjecture of Kato and Kuzumaki

Diego Izquierdo

Vol. 12 (2018), No. 2, 429–454

DOI: 10.2140/ant.2018.12.429

Abstract

In 1986, Kato and Kuzumaki stated several conjectures in order to give a diophantine characterization of cohomological dimension of fields in terms of projective hypersurfaces of small degree and Milnor $K$ -theory. We establish these conjectures for finite extensions of $ℂ (x_{1}, \dots, x_{n})$ and $ℂ (x_{1}, \dots, x_{n}) ((t))$ , and we prove new local-global principles over number fields and global fields of positive characteristic in the context of Kato and Kuzumaki’s conjectures.

Keywords

Cohomological dimension of fields, $C_i$ property, Milnor K-theory, Number fields, Function fields

Mathematical Subject Classification 2010

Primary: 11E76

Secondary: 12E25, 12E30, 14G27, 19D45, 19F99

Milestones

Received: 8 June 2017

Revised: 23 October 2017

Accepted: 18 December 2017

Published: 13 May 2018

Authors

Diego Izquierdo
Département de Mathématiques et Applications École Normale Supérieure - CNRS, PSL Research University Paris France

Vol. 12, No. 2, 2018