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By Bartolucci D.

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In general a mixed characteristic field is a. finite extension of a field like Q1,{{t1}} ... {{tm_1}} ([FV, sect. 5 Ch. II], for more details see [Zh]). It occurs that the Milnor K-groups are not the most suitable objects to be related with abelian extensions of an n-dimensional local field F. ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATTON FIELDS 49 It is more convenient to work with quotients of Milnor K-groups endowed with a special topology (A,,,, (F) is the intersection of all neighbourhoods of zero).

2. - Topology on the multiplicative group It is natural to expect compatibility of theories of a field and its residue field (lifting of extensions of the residue field as unramified extensions of the field and the border homomorphism in K-theory). Thus, a genuine topology on a multidimensional field has to take into account topologies of its residue fields. _1 (which is uniquely determined, see [FV, sect. 5 Ch. II]) when char(F) = 0. The ring 0 contains the set of canonical liftings R from ko, so called multiplicative representatives.

The ring 0 contains the set of canonical liftings R from ko, so called multiplicative representatives. g. [Ka3]). Then every element a E F* can be expanded as a convergent with respect to the just defined topology product a = t22t1 JJ(1 + i9 ,it2ti) with 0 54 0, Oi,4 E R, a1, a2 E Z. In the multidimensional case one can define topology by induction on dimension. Let F be an n-dimensional local field with char(kn_1) = p. Fix a lifting (and thus a set of representatives S) of kn_1 in F compatible with the residue morphism : ko -* 0o, the residues ti E kn_1, 1 < i < n - 1, of local parameters go to t, in F.

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A ''sup+ c inf'' inequality for Liouville-type equations with singular potentials by Bartolucci D.


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