Sur une conjecture de Kato et Kuzumaki concernant les hypersurfaces de Fano

TL;DR

This paper proves that p-adic and totally imaginary number fields satisfy a specific property related to expressing elements as products of norms from extensions where certain polynomials have zeros, confirming conjectures by Kato, Kuzumaki, and Ax.

Contribution

It establishes the C_1^1 property for p-adic and totally imaginary number fields and proves Ax's conjecture for fields with pro-p Galois groups.

Findings

p-adic fields satisfy the C_1^1 property

totally imaginary number fields satisfy the C_1^1 property

Ax's conjecture is confirmed for fields with pro-p Galois groups

Abstract

Nous montrons que les corps p-adiques et les corps de nombres totalement imaginaires v\'erifient la propri\'et\'e C_1^1 conjectur\'ee par Kato et Kuzumaki en 1986. Autrement dit, si k est l'un de ces corps et f(x_0, ..., x_n) est un polyn\^ome homog\`ene de degr\'e au plus n \`a coefficients dans k, tout \'el\'ement de k s'\'ecrit comme produit de normes depuis des extensions finies de k dans lesquelles f admet un z\'ero non trivial. Nous \'etablissons aussi la conjecture d'Ax sur les corps pseudo-alg\'ebriquement clos parfaits pour les corps dont le groupe de Galois absolu est un pro-p-groupe. ----- We prove that p-adic fields as well as totally imaginary number fields satisfy the C_1^1 property conjectured by Kato and Kuzumaki in 1986. In other words, if k denotes such a field and f(x_0, ..., x_n) is a homogeneous polynomial of degree at most n with coefficients in k, every element of k may be written as a product of norms from finite extensions of k in which f possesses a nontrivial zero. As a by-product of the proof, we also establish Ax's conjecture about perfect pseudo-algebraically closed fields for fields whose absolute Galois group is a pro-p-group.