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 ⋅ X1d + a2 ⋅ X2d + … + an⋅ Xnd ≡  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 = Zp)?

If R = Fp = 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 = Zp is the ring of p-adic integers (which is a generalization of Z/pαZ), and discuss a conjecture of Artin.