Norms on \(R[ X_{1}, \dots , X_r]\) which are multiplicative on \(R\)
\loadeufm The main motivation of the paper under review comes from the theory of local fields. By Galois theory in \({\mathbb C}_ p\), the completion of an algebraic closure \(\overline{\mathbb Q}_ p\) of the field of \(p\)-adic numbers \({\mathbb Q}_ p\), there is a natural correspondence between the closed subgroups of the Galois groups \(\roman{Gal}_{\text{cont}}({\mathbb C}_ p/{\mathbb Q}_ p)\) of continuous automorphisms of \({\mathbb C}_ p\) over \({\mathbb Q}_ p\) and the closed subfields of \({\mathbb C}_ p\). Now, for each closed subfield \(E\) of \({\mathbb C}_ p\), there exists an element \(T\) of \(E\) such that \(E\) is the topological closure of the ring \({\mathbb Q}_ p[T]\) in \({\mathbb C}_ p\), where \(T\) is transcendental over \({\mathbb Q}_ p\) when \(E\) is not a finite extension of \({\mathbb Q}_ p\). Although all these rings are ring isomorphic, what is very different for various transcendental elements \(T\) over \({\mathbb Q}_ p\) is the valuation on \({\mathbb C}_ p\) restricted to \({\mathbb Q}_ p[T]\). That is, this variety of closed subfields of \({\mathbb C}_ p\) is determined by these extensions of the \(p\)-adic valuation from \({\mathbb Q}_ p\) to the ring \({\mathbb Q}_ p[X]\) of polynomials of one variable over \({\mathbb Q}_ p\). There are some other objects obtained in a similar way: the rings \(B_{dR}^ +\) introduced by \textit{J.-M. Fontaine} [Ann. Math. (2) 115, 529--577 (1982; Zbl 0544.14016)], the rings \(B_ n:=B_{dR}^ +/I^ n\) where \(I\) is the maximal ideal of \(B_{dR}^ +\), have generating elements over \({\mathbb Q}_ p\), so these rings can also be obtained as completions of \({\mathbb Q}_ p[X]\) with respect to some norms which extend the \(p\)-adic absolute value from \({\mathbb Q}_ p\). Other motivation comes from the structure of some spaces of functions, which were introduced to define some types of convergence for different sequences of real functions, which are not uniformly convergent. These spaces are completions of \({\mathbb R}[X]\) with respect to some norms which extend the ordinary absolute value on \({\mathbb R}\). Thus one could gain new insight into these problems of understanding the structure of these rings and fields analyzing and classifying the corresponding valuations defined on appropriate polynomial rings. This paper takes a more general point of view. The authors fix a valued ring \(R\) and discuss some properties of norm extensions to polynomial rings \(R[X_ 1,\ldots, X_ r]\) in several variables, where \(R\) is a commutative ring with identity. Given a non-trivial absolute value on \(R\), the authors study \(R\)-algebras norms \(\| \cdot\| \) which extend \(| \cdot | \). In particular it is proved that when \(R\) is a field, two norms are equivalent if and only if they produce isomorphic completions.
scientific article; zbMATH DE number 5285866