Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken

1989 ◽  
Vol 54 (1) ◽  
pp. 234-263 ◽  
Author(s):  
H. Luckhardt
Keyword(s):  
Continued Fraction ◽  
Diophantine Equations ◽  
Growth Conditions ◽  
Number Of Solutions ◽  
Exponential Bound ◽  
Low Degree ◽  
Diophantine Approximations ◽  
Finiteness Theorems ◽  
Fourth Root ◽  
Simple Growth

AbstractA previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs ofΣ2-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε−2d2) in place of exp(285ε−2d2) given by Davenport and Roth in 1955, whereαis (real) algebraic of degreed≥ 2 and ∣α−pq−1∣ <q−2−ε. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof arepolynomial of low degree(inε−1and logd). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), log logqν, <Cα, εν5/6+ε, whereqνare the denominators of the convergents to the continued fraction ofα.

The paucity problem for simultaneous quadratic and biquadratic equations

1999 ◽  
Vol 126 (2) ◽  
pp. 209-221 ◽  
Author(s):  
W. Y. TSUI ◽  
T. D. WOOLEY
Keyword(s):  
Lower Bounds ◽  
Diophantine Equations ◽  
Upper And Lower Bounds ◽  
Current Interest ◽  
Sieve Methods ◽  
Number Of Solutions ◽  
Simultaneous Diophantine Equations ◽  
History Of

The problem of constructing non-diagonal solutions to systems of symmetric diagonal equations has attracted intense investigation for centuries (see [5, 6] for a history of such problems) and remains a topic of current interest (see, for example, [2–4]). In contrast, the problem of bounding the number of such non-diagonal solutions has commanded attention only comparatively recently, the first non-trivial estimates having been obtained around thirty years ago through the sieve methods applied by Hooley [10, 11] and Greaves [7] in their investigations concerning sums of two kth powers. As a further contribution to the problem of establishing the paucity of non-diagonal solutions in certain systems of diagonal diophantine equations, in this paper we bound the number of non-diagonal solutions of a system of simultaneous quadratic and biquadratic equations. Let S(P) denote the number of solutions of the simultaneous diophantine equationsformula herewith 0[les ]xi, yi[les ]P(1[les ]i[les ]3), and let T(P) denote the corresponding number of solutions with (x1, x2, x3) a permutation of (y1, y2, y3). In Section 4 below we establish the upper and lower bounds for S(P)−T(P) contained in the following theorem.

MODUS OPERANDIIN ASSESSING BIOMASS AND CARBON IN RUBBER PLANTATIONS UNDER VARYING CLIMATIC CONDITIONS

2013 ◽  
Vol 50 (1) ◽  
pp. 40-58 ◽  
Author(s):  
E. S. MUNASINGHE ◽  
V. H. L. RODRIGO ◽  
U. A. D. P. GUNAWARDENA
Keyword(s):  
Rubber Tree ◽  
Rural Poverty ◽  
Growth Conditions ◽  
Climatic Conditions ◽  
Percentage Error ◽  
Carbon Trading ◽  
Average Growth ◽  
Allometric Models ◽  
The Mean ◽  
Simple Growth

SUMMARYIn addition to latex and timber, the rubber tree is useful in the alleviation of rural poverty and also in the mitigation of climate change through fixing atmospheric CO2as biomass. For developing any rubber-based carbon projects, protocols for quantifying biomass and carbon fixed are required. In this context, the present study was aimed at building up allometric models using simple growth indicators (i.e. tree diameter and total height) to assess the timber, biomass and carbon in rubber trees and also to quantify their ontogenetic variation under average growth conditions in two major climatic regimes (i.e. wet and intermediate) of Sri Lanka. All models developed were in the accuracy level of over 88%. The mean absolute percentage error in the validation of allometric models was only 12.9% for timber and less than 5% for biomass and carbon. Under average growth conditions, 1 ha of rubber could produce 208 m3timber, 191 MT biomass and fix 78 MT carbon during its 30-year lifespan in the wet zone and ca. 16% lesser values in the intermediate zone. The applicability of the findings in carbon trading is discussed.

On Diophantine approximations of the solutions of q-functional equations

2002 ◽  
Vol 132 (3) ◽  
pp. 639-659 ◽  
Author(s):  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Number Fields ◽  
Growth Conditions ◽  
Linear Forms ◽  
Dimension Estimate ◽  
Linearly Independent ◽  
Diophantine Approximations ◽  
Image Position

Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

scholarly journals A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S &sube; {xi &middot; xj = xk : i, j, k &isin; {1, . . . , n}} &cup; {xi + 1 = xk : i, k &isin;{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn &le; f (2n). We prove:&nbsp;&nbsp; (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M &sube; N has a finite-fold Diophantine representation, then M is computable.

scholarly journals Generalized Ehrhart polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sheng Chen ◽  
Nan Li ◽  
Steven V Sam
Keyword(s):  
Diophantine Equations ◽  
Rational Functions ◽  
Lattice Points ◽  
Classical Theorem ◽  
Number Of Solutions ◽  
Ehrhart Polynomials ◽  
Linear Diophantine Equations ◽  
International Audience

International audience Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related. Soit $P$ un polytope avec sommets rationelles. Un théorème classique des Ehrhart déclare que le nombre de points du réseau dans les dilatations $P(n) = nP$ est un quasi-polynôme en $n$. Nous généralisons ce théorème en permettant à des sommets de $P(n)$ comme arbitraire fonctions rationnelles en $n$. Dans ce cas, nous prouvons que le nombre de points du réseau en $P(n)$ est une quasi-polynôme pour $n$ assez grand. Notre travail a été motivée par une conjecture d'Ehrhart sur le nombre de solutions à linéaire paramétrée Diophantine équations dont les coefficients sont des polyômes en $n$, et nous expliquer comment ces deux problèmes sont liés.

scholarly journals Use of Growth-Rate/Temperature-Gradient Charts for Defect Engineering in Crystal Growth from the Melt

Crystals ◽  
2020 ◽  
Vol 10 (10) ◽  
pp. 909
Author(s):  
Thierry Duffar
Keyword(s):  
Growth Rate ◽  
Crystal Growth ◽  
Temperature Gradient ◽  
Growth Conditions ◽  
Industrial Applications ◽  
Physical Models ◽  
Production Yield ◽  
Defect Engineering ◽  
Pedagogic Tool ◽  
Simple Growth

As the requirements in terms of crystal defect/quality and production yield are generally contradictory, it is necessary to develop methods in order to find the best compromise for the growth conditions of a given crystal. Simple growth-rate/temperature-gradient charts are a possible tool in this respect. After the recall of the classical analytical equations useful for describing the process and defect engineering, a simple pedagogic case explains the building and use of such charts. The more complex application to the directional casting of photovoltaic Si necessitated the development of new physical models for twinning and equiaxed growth. This allowed plotting charts that proved useful for industrial applications. The conclusions discuss the drawbacks and advantages of the method. It finally proves to be a pedagogic tool for teaching crystal growth engineering.

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