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Authors
Krzysztof Krupinski
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We prove that every non-trivial valuation on an infinite superrosy field of positive characteristic has divisible value group and algebraically closed residue field. In fact, we prove the following more general result. Let $K$ be a field such that for every finite extension $L$ of $K$ and for every natural number $n>0$ the index $[L^*:(L^*)^n]$ is finite and, if $char(K)=p>0$ and $f: L to L$ is given by $f(x)=x^p-x$, the index $[L^+:f[L]]$ is also finite. Then either there is a non-trivial definable valuation on $K$, or every non-trivial valuation on $K$ has divisible value group and, if $char(K)>0$, it has algebraically closed residue field. In the zero characteristic case, we get some partial results of this kind. We also notice that minimal fields have the property that every non-trivial valuation has divisible value group and algebraically closed residue field.
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We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.
We give a valuation theoretic characterization for a real closed field to be recursively saturated. Our result extends the characterization of Harnik and Ressayre cite{hr} for a divisible ordered abelian group to be recursively saturated.
Suppose that (K, $ u$) is a valued field, f (z) $in$ K[z] is a unitary and irreducible polynomial and (L, $omega$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $ u$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $omega$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $omega$ as an extension of the associated graded ring of A for the filtration defined by $ u$. In particular we give an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr $omega$ A[z]/(f (z)) generate it as a gr $ u$ A-algebra as well as the relations between them. We also work out the interactions of our method of computation with phenomena which complicate the study of ramification and local uniformization in positive characteristic , such as the non tameness and the defect of an extension. For valuations of rank one in a separable extension of valued fields (K, $ u$) $subset$ (L, $omega$) as above our algorithm produces a generating sequence in a local birational extension A1 of A dominated by $ u$ if and only if there is no defect. In this case, gr $omega$ A1[z]/(f (z)) is a finitely presented gr $ u$ A1-module. This is an improved version, thanks to a referees remarks.
In this paper, we study factorizations in the additive monoids of positive algebraic valuations $mathbb{N}_0[alpha]$ of the semiring of polynomials $mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when $mathbb{N}_0[alpha]$ is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid $mathbb{N}_0[alpha]$, the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for $mathbb{N}_0[alpha]$ to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of $alpha$. The properties of being finitely generated, half-factorial, and other-half-factorial are also investigated along the way.
We give two concrete examples of continuous valuations on dcpos to separate minimal valuations, point-continuous valuations and continuous valuations: (1) Let $mathcal J$ be the Johnstones non-sober dcpo, and $mu$ be the continuous valuation on $mathcal J$ with $mu(U) =1$ for nonempty Scott opens $U$ and $mu(U) = 0$ for $U=emptyset$. Then $mu$ is a point-continuous valuation on $mathcal J$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $mathbb R_{l}$. Its restriction to the open subsets of $mathbb R_{l}$ is a continuous valuation $lambda$. Then its image valuation $overlinelambda$ through the embedding of $mathbb R_{l}$ into its Smyth powerdomain $mathcal Qmathbb R_{l}$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $overlinelambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpos.
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