On a conjecture of Kato and Kuzumaki

Diego Izquierdo

doi:10.2140/ant.2018.12.429

On a conjecture of Kato and Kuzumaki

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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(x₁,…, x_n) and ℂ(x₁,…, 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.

Original language	English
Pages (from-to)	429-454
Number of pages	26
Journal	Algebra and Number Theory
Volume	12
Issue number	2
DOIs	https://doi.org/10.2140/ant.2018.12.429
Publication status	Published - 1 Jan 2018

Keywords

C property
Cohomological dimension of fields
Function fields
Milnor K-theory
Number fields

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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(x1,…, xn) and ℂ(x1,…, xn)((t)), and we prove new local-global principles over number fields and global fields of positive characteristic in the context of Kato and Kuzumaki{\textquoteright}s conjectures.",

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AB - 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(x1,…, xn) and ℂ(x1,…, xn)((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.

KW - C property

KW - Cohomological dimension of fields

KW - Function fields

KW - Milnor K-theory

KW - Number fields

U2 - 10.2140/ant.2018.12.429

DO - 10.2140/ant.2018.12.429

M3 - Article

AN - SCOPUS:85047237271

SN - 1937-0652

VL - 12

SP - 429

EP - 454

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 2

ER -

On a conjecture of Kato and Kuzumaki

Abstract

Keywords

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