1 Henselian extensions Application of ZMT

In all this paper A will be a local ring, with a detachable maximal ideal M. We let k be the residue field A/M. If we have such a local ring A, M it is convenient to think of the elements of M as “infinitesimal”, whereas the elements of A × are the ones that are observationally different from 0. (The introduction of [8] is helpful there.) We shall look at a polynomial system f1(x1,..., xn) =... = fn(x1,..., xn) = 0 (∗) which has a simple zero at (0,..., 0) residually: we have not only fi(0,..., 0) = 0 residually but also the Jacobian of this system J(0,..., 0) is in A ×. We are going to associate, in an explicit way, to such a system a unitary polynomial f of degre m which is of the form X m−1 (X − 1) residually. To this polynomial we can associate the extension Af of A obtained by forcing f(z) = 0 and inverting all elements g(z) such that g(1) ∈ A ×. Intuitively we have added a root of f which is infinitely close to 1. The extension Af is called a simple Hensel extension of A. One can show that Af is a local ring and we have a local embedding of A into Af, the maximal ideal Mf being the set of elements h(z)/g(z) such that h(1) ∈ M [1]. (This is actually rather direct since f is unitary.) For instance we hav