Application of ZMT
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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 Xm−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 have z − 1 ∈ Mf and this expresses that z is infinitely close to 1. The polynomial f will be such that in Af there is a solution (x1, . . . , xn) of the system (∗) where all x1, . . . , xn are in Mf . Thus we have found a local extension of A in which the system (∗) has a solution “infinitely close” to 0. A unitary polynomial which is of degre m and of the form Xm−1(X − 1) residually is called a special polynomial. Notice that if f is a special polynomial we always have f(1) = 0 and f ′(1) = 1 residually. Notice also that z is a unit of Af . We call such an element a special unit. We can summarise this discussion by the following result.
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تاریخ انتشار 2006