Norm form equations. III: positive characteristic

1986 ◽  
Vol 99 (3) ◽  
pp. 409-423 ◽  
Author(s):  
R. C. Mason
Keyword(s):  
Complete Resolution ◽  
Positive Characteristic ◽  
Approximation Theorem ◽  
Function Fields ◽  
Zero Characteristic ◽  
Norm Form ◽  
Starting Point ◽  
Diophantine Problems ◽  
General Norm ◽  
Image Position

This paper aims to provide a complete resolution of the general norm form equation over function fields of positive characteristic. In a previous paper [4] we studied norm forms in the simpler case of zero characteristic; that study forms the starting point for the present investigations. Diophantine problems over function fields of positive characteristic were first investigated by Armitage in 1968 [1], who clamied to have established an analogue of the Thue–Siegel–Roth–Uchiyama theorem for such fields. This claim was refuted by Osgood in 1975 [6], who also derived a correct analogue of Thue's approximation theorem. in 1983 a different attack was made on Diophantine problems over function fields, the principal weapon being a bound [2] for the heights of the solutions of the unit equation

Bernstein Power Series

1964 ◽  
Vol 16 ◽  
pp. 241-252 ◽  
Author(s):  
E. W. Cheney ◽  
A. Sharma
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Approximation Theorem ◽  
Bernstein Polynomials ◽  
Starting Point ◽  
Image Position ◽  
Weierstrass Approximation Theorem

In Bernstein's proof of the Weierstrass Approximation Theorem, the polynomialsare constructed in correspondence with a function f ∊ C [0, 1] and are shown to converge uniformly to f. These Bernstein polynomials have been the starting point of many investigations, and a number of generalizations of them have appeared. It is our purpose here to consider several generalizations in the form of infinite series and to establish some of their properties.

Vanishing sums in function fields

1986 ◽  
Vol 100 (3) ◽  
pp. 427-434 ◽  
Author(s):  
W. D. Brownawell ◽  
D. W. Masser
Keyword(s):  
Projective Space ◽  
Function Field ◽  
Function Fields ◽  
Order Group ◽  
Zero Characteristic ◽  
Sum Formula ◽  
Image Position

Let k be a field of zero characteristic, and let F be a function field over k of genus g. We normalize each valuation v on F so that its order group consists of all rational integers, and for elements u1, …, un of F, not all zero, we define the (projective) height asThe sum formula on F shows that this is really a height on the projective space .

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals Function fields in positive characteristic: Expansions and Cobham's theorem

2008 ◽  
Vol 319 (6) ◽  
pp. 2337-2350 ◽  
Author(s):  
Boris Adamczewski ◽  
Jason Bell
Keyword(s):  
Positive Characteristic ◽  
Function Fields

On a set theory of bernays

1967 ◽  
Vol 32 (3) ◽  
pp. 319-321 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Free Variable ◽  
Reflection Principle ◽  
Formal Definition ◽  
Von Neumann ◽  
Starting Point ◽  
Definition Of ◽  
Image Position

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.

RESIDUAL PROPERTIES OF FREE PRO-P GROUPS

2001 ◽  
Vol 33 (5) ◽  
pp. 578-582 ◽  
Author(s):  
YIFTACH BARNEA
Keyword(s):  
Positive Characteristic ◽  
Congruence Subgroup ◽  
Probabilistic Method ◽  
Algebraic Groups ◽  
Local Fields ◽  
Random Generation ◽  
Zero Characteristic ◽  
Residual Properties ◽  
Characteristic Case ◽  
Lie Methods

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

Sliding in rough convex bowls

2014 ◽  
Vol 98 (541) ◽  
pp. 8-23
Author(s):  
Mark Hennings ◽  
Jon Ingram
Keyword(s):  
Coefficient Of Friction ◽  
Initial Speed ◽  
Interior Surface ◽  
Starting Point ◽  
The Coefficient Of Friction ◽  
Braking Distance ◽  
Image Position ◽  
Elementary Fact

It is an elementary fact that a particle sliding on a rough horizontal table experiences a deceleration of magnitude μg, where μ is the coefficient of friction between the particle and the table. Thus, if the initial speed u of the particle and the braking distance d are both known, then μ can be determined by the formulaThe idea of using braking distances to determine the coefficient of friction led the second author to ask the following question:A particle slides on the interior surface of a rough hemispherical bowl starting from rest. If the starting point of the particle and the point at which it comes to instantaneous rest after its slide are both known, is it possible to determine the coefficient of friction between the particle and the bowl?

scholarly journals MINOR ARC MOMENTS OF WEYL SUMS

2012 ◽  
Vol 55 (1) ◽  
pp. 97-113 ◽  
Author(s):  
M. P. HARVEY
Keyword(s):  
Diophantine Equations ◽  
Diagonal Form ◽  
Norm Form ◽  
Goldbach Problem ◽  
Weyl Sum ◽  
Diophantine Problems

AbstractWe obtain an improved bound for the 2k-th moment of a degree k Weyl sum, restricted to a set of minor arcs, when k is small. We then present some applications of this bound to some Diophantine problems, including a case of the Waring–Goldbach problem, and a particular family of Diophantine equations defined as the sum of a norm form and a diagonal form.

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