Abstract 7

Can we find a number n=n(d,r) such that every equation of the type

a₁ ⋅ X₁^d + a₂ ⋅ X₂^d + … + a_n⋅ X_n^d ≡ 0 (mod r)

has a non-trivial solution?

More generally, can we find a number N=N(d) such that every homogeneous polynomial f of degree d, in at least N variables, has a non-trivial zero in a given ring R (for example with R = Z, R = Z/rZ or R = Z_p)?

If R = F_p = Z/pZ (where p is a prime), then the theorem of Chevalley-Warning tells us that it is true if we take N(d) = d+1.

In this talk, we will study the case where R = Z_p is the ring of p-adic integers (which is a generalization of Z/p^αZ), and discuss a conjecture of Artin.

	Nicolas Bartholdi
	The diophantine dimension of the p-adic fields

Can we find a number n=n(d,r) such that every equation of the type a₁ ⋅ X₁^d + a₂ ⋅ X₂^d + … + a_n⋅ X_n^d ≡ 0 (mod r) has a non-trivial solution? More generally, can we find a number N=N(d) such that every homogeneous polynomial f of degree d, in at least N variables, has a non-trivial zero in a given ring R (for example with R = Z, R = Z/rZ or R = Z_p)? If R = F_p = Z/pZ (where p is a prime), then the theorem of Chevalley-Warning tells us that it is true if we take N(d) = d+1. In this talk, we will study the case where R = Z_p is the ring of p-adic integers (which is a generalization of Z/p^αZ), and discuss a conjecture of Artin.