On a valuation of rational subsets of Dédié à Jean Berstel (original) (raw)

On a valuation of rational subsets of Z k Dédié à Jean Berstel

Theoretical Computer Science, 2003

A well-known construction associates to each rational subset of N k a rational function in k commuting variables. We extend this construction to rational subsets of Z k . As a consequence, we derive a multivariate generalization of Popoviciu's theorem and a classical valuation on rational polyhedra.

Valuations on lattice polytopes

Advances in Mathematics, 2009

Let L be a lattice (that is, a Z-module of finite rank), and let L = P(L) denote the family of convex polytopes with vertices in L; here, convexity refers to the underlying rational vector space V = Q ⊗ L. In this paper it is shown that any valuation on L satisfies the inclusion-exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group L = P(L) for valuations on L can be identified with its characteristic function in V. In fact, the same arguments are shown to apply to P(M), when M is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra.

The Growth of Valuations on Rational Function Fields in Two Variables

2004

Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k[x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q --+ k that has Noetherian (i.e., reverse well-ordered) support. Each

A construction for a class of valuations of the field k(X1,…,Xd,Y) with large value group

Journal of Algebra, 2008

Given any algebraically closed field k of characteristic zero and any totally ordered abelian group G of rational rank less than or equal to d, we construct a valuation of the field k(X 1 ,. .. , X d , Y) with value group G. In the case of rational rank equal to d this valuation is induced by a formal fractional power series parametrization of a transcendental hypersurface in affine (d + 1)-space which is naturally approximated by a sequence of quasi-ordinary hypersurfaces. The value semigroup ν(k[X, Y ] \ {0}) is the direct limit of the semigroups associated to these quasi-ordinary hypersurfaces.

The Growth of Valuations on Rational Function Fields

2001

A brief introduction to valuations on lattice polytopes

2019

These notes are based on a five-lecture summer school course given by the author at the “Summer Workshop on Lattice Polytopes” at Osaka University in 2018. We give a short introduction to the theory of valuations on lattice polytopes. Valuations are a classical topic in convex geometry. The volume plays an important role in many structural results, such as Hadwiger’s famous characterization of continuous, rigid-motion invariant valuations on convex bodies. Valuations whose domain is restricted to lattice polytopes are less well-studied. The Betke-Kneser Theorem establishes a fascinating discrete analog of Hadwiger’s Theorem for lattice-invariant valuations on lattice polytopes in which the number of lattice points — the discrete volume — plays a fundamental role. From there, we explore striking parallels, analogies and also differences between the world of valuations on convex bodies and those on lattice polytopes with a focus on positivity questions and links to Ehrhart theory.

Valuation leveling on rational function fields

Since a new field of research in the theory of valuations has been opened by the notion of symmetric extensions of a valuation on a field K to K(X1,…, Xn), with respect to the indeterminates X1,…, Xn, it makes sense to look for applications of this theory in dealing with arbitrary valuations on rational function fields. This paper investigates the conditions and methods of transforming asymmetric valuations on rational function fields into symmetric ones-procedure that will be called valuation leveling-the motivation being the opportunity of applying the specific results from the theory of symmetric valuations in the general case. 1. Background Given K a field and v a (Krull) valuation on K, we will denote by vK the value group, by Kv the residue field of v on K, by va the value of an element a  K and by av  Kv its residue. We will use the classical additive notation for v, that is assuming the ultrametric triangle law as v(a + b) ≥ inf (a, b). Given v and u two valuations on K, we will say that v is equivalent to u and write v  u, if there exists an isomorphism of ordered groups j : vK  uK such that we get u = jv. Let L/K be an extension of fields. We will call a valuation u on L an extension of v if u(x) = v(x) for all x in K and, in this case, we will canonically identify Kv with a subfield of Lu and vK with a subgroup of uL. If we choose L

Desity of rational points on

2016

The purpose of this paper is to combine classical methods from transcendental number theory with the technique of restriction to real scalars. We develop a conceptual approach relating transcendence properties of algebraic groups to results about the existence of homomorphisms to group varieties over real fields. Our approach gives a new perspective on Mazur's conjecture on the topology of rational points. We shall reformulate and generalize Mazur's problem in the light of transcendence theory and shall derive conclusions in the direction of the conjecture. Next to these new theoretical insights, the aim of our application motivated Ansatz was to improve classical results of transcendence, of algebraic independence in small transcendence degree and of linear independence of algebraic logarithms. Thirty new corollaries, most of which are generalizations of popular theorems, are stated in the seventh chapter. For example we shall prove: Let a 1 , a 2 , a 3 be three linearly independent complex numbers, let ℘(z) be a Weierstraß function with algebraic invariants and let b be a non-zero complex numbers. If the four numbers satisfy certain hypotheses, then one among the six numbers ℘(a j), e ba j is transcendental. Let ℘(z) be a Weierstraß function with algebraic invariants and complex multiplication by √ −d for a square-free integer d > 1. If ℘(ω) is defined and algebraic, then either ω/|ω| is algebraic or ω/|ω| and ℘(iω) are algebraically independent. Let ℘(z) be a Weierstraß function with algebraic invariants and lattice Λ and let ω be a complex number such that ℘(ω) is defined and algebraic. Then rω is transcendental for each r > 0 or rω ∈ Λ for some r > 0. This elaborated long version of our work is essentially self-contained and should be admissible for master students specializing in transcendental number theory and arithmetic geometry.

On a valuation of rational subsets of Dédié à Jean Berstel (original) (raw)

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