#### Vol. 12, No. 2, 2018

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On a conjecture of Kato and Kuzumaki

### Diego Izquierdo

Vol. 12 (2018), No. 2, 429–454
##### 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 $ℂ\left({x}_{1},\dots ,{x}_{n}\right)$ and $ℂ\left({x}_{1},\dots ,{x}_{n}\right)\left(\left(t\right)\right)$, 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