Factorial henselian rings with non-factorial completions

Newton Polyhedra and Factorial Rings

be a Laurent polynomial over a field k. The convex hull of the set {(ii,. . . , i,)~ Z” : a i,...i. f 0) in R” is called the Newton polyhedron of F and will be denoted by 4~. Following Hovanskii [S] we say that F is nondegenerate if for any face d’ of AF the hypersurface C~i,...,.in)Ed,Ui,...i,Xill * * * xi = 0 is non-singular in F*” (Eis the algebraic closure of the field k). The main result o...

Factorial Rings and Diagonal Reduction of Matrices

The class of Bézout factorial rings is introduced and characterized. Using the factorial properties of such a ring R, and given a n×m matrix A over R, we find P ∈ GL(n, R) and Q ∈ GL(m, R) such that PAQ is diagonal with every element in the diagonal dividing the following one. Key-words: Ring, Bézout, principal, factorization, reduction of matrices.

Factorial and Noetherian Subrings of Power Series Rings

Let F be a field. We show that certain subrings contained between the polynomial ring F [X] = F [X1, · · · , Xn] and the power series ring F [X][[Y ]] = F [X1, · · · , Xn][[Y ]] have Weierstrass Factorization, which allows us to deduce both unique factorization and the Noetherian property. These intermediate subrings are obtained from elements of F [X][[Y ]] by bounding their total X-degree abo...

Definable Henselian Valuation Rings

We give model theoretic criteria for the existence of∃∀ and∀∃formulas in the ring language to define uniformly the valuation rings O of models (K,O) of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings O of valued henselian fields (K,O) whose residue class field k is finite, pseudofinite,...

Factorial Design