scholarly journals APPROXIMATIONS FONCTIONNELLES DES COURBES DES ESPACES PROJECTIFS

2011 ◽  
Vol 07 (05) ◽  
pp. 1195-1215 ◽  
Author(s):  
PATRICE PHILIPPON
Keyword(s):  
Diophantine Approximation ◽  
Function Field ◽  
Algebraic Closure ◽  
Algebraic Independence ◽  
Projective Spaces ◽  
Higher Dimension ◽  
Algebraic Numbers ◽  
Flexible Approach ◽  
Algebraic Approximation ◽  
Multiplicity Estimates

Algebraic approximation to points in projective spaces offers a new and more flexible approach to algebraic independence theory. When working over the field of algebraic numbers, it leads to open conjectures in higher dimension extending known results in Diophantine approximation. We show here that over the algebraic closure of a function field in one variable, the analog of these conjectures is true. We also derive transfer lemmas which have applications in the study of multiplicity estimates, for example.

Sets with large intersection and ubiquity

2008 ◽  
Vol 144 (1) ◽  
pp. 119-144 ◽  
Author(s):  
ARNAUD DURAND
Keyword(s):  
Diophantine Approximation ◽  
Nondecreasing Function ◽  
Gauge Function ◽  
Algebraic Numbers ◽  
Positive Real ◽  
Integer Polynomials ◽  
Large Intersection ◽  
Image Position ◽  
Full Lebesgue Measure ◽  
Arbitrary Norm

AbstractA central problem motivated by Diophantine approximation is to determine the size properties of subsets of$\R^d$ ($d\in\N$)of the formwhere ‖⋅‖ denotes an arbitrary norm,Ia denumerable set, (xi,ri)i∈ Ia family of elements of$\R^d\$× (0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that ifFId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subsetVof$\R^d\$, the setFϕbelongs to a class Gh(V) of sets with large intersection inVwith respect to a given gauge functionh. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorffg-measure for every gauge functiongwhich increases faster thanhnear zero. In particular, this yields a sufficient condition on a gauge functiongsuch that a given countable intersection of sets of the formFϕhas infinite Hausdorffg-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequenceψof positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that areψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.

scholarly journals Projective metric number theory

2016 ◽  
Vol 2016 (712) ◽  
Author(s):  
Anish Ghosh ◽  
Alan Haynes
Keyword(s):  
Number Theory ◽  
Projective Space ◽  
Diophantine Approximation ◽  
Higher Dimension ◽  
Probabilistic Theory ◽  
Metric Number Theory ◽  
Projective Metric

AbstractIn this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of ℚ. Using the projective metric studied in [Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 2, 211–248] we prove the analogue of Khintchine's theorem in projective space. For finite places and in higher dimension, we are able to completely remove the condition of monotonicity and establish the analogue of the Duffin–Schaeffer conjecture.

scholarly journals Algebraic independence properties of the Fredholm series

1978 ◽  
Vol 26 (1) ◽  
pp. 31-45 ◽  
Author(s):  
J. H. Loxton ◽  
A. J. van der Poorten
Keyword(s):  
Algebraic Independence ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Numbers ◽  
Independence Properties ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

COUNTING MULTISECTIONS IN CONIC BUNDLES OVER A CURVE DEFINED OVER 𝔽q

2011 ◽  
Vol 07 (06) ◽  
pp. 1663-1680
Author(s):  
SEYFI TÜRKELLI
Keyword(s):  
Function Field ◽  
Algebraic Numbers ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Special Case

For a given conic bundle X over a curve C defined over 𝔽q, we count irreducible branch covers of C in X of degree d and height e ≫ 1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field 𝔽q(C).

scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

Counting points of given height that generate a quadratic extension of a function field

2015 ◽  
Vol 11 (02) ◽  
pp. 569-592 ◽  
Author(s):  
David Kettlestrings ◽  
Jeffrey Lin Thunder
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Function Field ◽  
Asymptotic Estimate ◽  
Algebraic Closure ◽  
Quadratic Extension ◽  
Rational Functions ◽  
Rational Numbers ◽  
Algebraic Extension ◽  
Counting Points

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

Defining transcendentals in function fields

2002 ◽  
Vol 67 (3) ◽  
pp. 947-956 ◽  
Author(s):  
Jochen Koenigsmann
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Algebraic Closure ◽  
Rational Function Field ◽  
First Order

AbstractGiven any field K, there is a function field F/K in one variable containing definable transcendental over K, i.e., elements in F / K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t).For the proof, diophantine ∅-definability ofK in F is established for any function field F/K in one variable, provided K is large, or K× /(K×)n is finite for some integer n > 1 coprime to char K.

Algebraic independence

1981 ◽  
Vol 46 (2) ◽  
pp. 377-384 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Algebraic Closure ◽  
Algebraic Independence ◽  
Independent Set ◽  
Independent Sets ◽  
Large Cardinal ◽  
Similarity Type ◽  
Limit Cardinal ◽  
Free Set ◽  
Infinite Set ◽  
The Universe

This paper is concerned with algebraic independence in structures that are relatively simple for their size. It is shown that for κ a limit cardinal, if a structure of power at least κ is ∞ω-equivalent to a structure of power less than κ, then must contain an infinite set of algebraically independent elements. The same method of proof yields the fact that if σ is an Lω1ω-sentence (not necessarily complete) and σ has a model of power ℵω then some model of σ contains an infinite algebraically independent set.All structures are assumed to be of countable similarity type. Letters , etc. will be used to denote either a structure or the universe of the structure. If X ⊆ , the algebraic closure of X (in ), denoted by Cl(X), is the union of all finite sets that are weakly definable (in ) by Lωω-formulas with parameters from X. A set S is algebraically independent if for each a in S, a ∉ Cl(S – {a}). An algebraically independent set is sometimes called a “free” set (in [3] and [4], for example).It is known (see [5]) that any structure of power ℵn must have a set of n algebraically independent elements, and there are structures of power ℵn with no independent set of size n + 1. In power ℵω every structure will have arbitrarily large finite algebraically independent sets. However, it is consistent with ZFC that some models of power ℵω do not have any infinite algebraically independent set. Devlin [4] showed that if V = L, then for any cardinal κ, if every structure of power κ has an infinite algebraically independent set, then κ has a certain large cardinal property that ℵω can never possess.

scholarly journals Schlanke Körper (Slim fields)

2010 ◽  
Vol 75 (2) ◽  
pp. 481-500 ◽  
Author(s):  
Markus Junker ◽  
Jochen Koenigsmann
Keyword(s):  
Algebraic Closure ◽  
Algebraic Independence ◽  
Valued Fields ◽  
Henselian Valued Fields ◽  
Relative Field

AbstractWe examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

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