scholarly journals On real algebraic numbers in which the derivative of their minimal polynomial is small

2021 ◽  
Vol 57 (2) ◽  
pp. 135-147
Author(s):  
D. V. Koleda
Keyword(s):  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Numbers ◽  
Absolute Value ◽  
Positive Degree ◽  
Integer Polynomials ◽  
The Absolute

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

scholarly journals Perron units which are not Mahler measures

1986 ◽  
Vol 6 (4) ◽  
pp. 485-488 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Unit Circle ◽  
Ergodic Theory ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Mahler Measure ◽  
Absolute Value ◽  
Perron Number ◽  
The Absolute

AbstractThe Mahler measure M(α) of an algebraic integer α is the product of the absolute value of the conjugates of α which lie outside the unit circle. The quantity log M(α) occurs in ergodic theory as the entropy of an endomorphism of the torus. Adler and Marcus showed that if β = M(α) then β is a Perron number which is a unit if α is a unit. They asked whether the Perron number β whose minimal polynomial is tm −t −1 is the measure of any algebraic integer. We show here that the answer is negative for all m > 3.

A Note on Integer Symmetric Matrices and Mahler's Measure

2008 ◽  
Vol 51 (1) ◽  
pp. 57-59 ◽  
Author(s):  
Edward Dobrowolski
Keyword(s):  
Lower Bound ◽  
Minimal Polynomial ◽  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Absolute Value ◽  
The Absolute

AbstractWe find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.

Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

1976 ◽  
Vol 15 (1) ◽  
pp. 33-57 ◽  
Author(s):  
A.J. van der Poorten ◽  
J.H. Loxton
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
Computable Bound ◽  
Image Position ◽  
Detailed Computation

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

scholarly journals Counting real algebraic numbers with bounded derivative of minimal polynomial

2019 ◽  
Vol 15 (10) ◽  
pp. 2223-2239
Author(s):  
Alexey Kudin ◽  
Denis Vasilyev
Keyword(s):  
Diophantine Approximation ◽  
Upper Bound ◽  
Minimal Polynomial ◽  
Algebraic Numbers ◽  
Fixed Degree ◽  
Bounded Degree ◽  
Absolute Value ◽  
Polynomial Formula ◽  
Integer Polynomial ◽  
Bounded Derivative

In this paper, we consider the problem of counting algebraic numbers [Formula: see text] of fixed degree [Formula: see text] and bounded height [Formula: see text] such that the derivative of the minimal polynomial [Formula: see text] of [Formula: see text] is bounded, [Formula: see text]. This problem has many applications to the problems of metric theory of Diophantine approximation. We prove that the number of [Formula: see text] defined above on the interval [Formula: see text] does not exceed [Formula: see text] for [Formula: see text] and [Formula: see text]. Our result is based on an improvement to a lemma from Gelfond’s monograph “Transcendental and algebraic numbers”. Given an integer polynomial small enough in some point, the lemma provides an upper bound for the absolute value of its irreducible divisor. We obtain a stronger estimate which holds in real points located far enough from all algebraic numbers of bounded degree and height. This is done by considering the resultant of two polynomials represented as the determinant of the Sylvester matrix for the shifted counterparts.

scholarly journals Incentive Function of Audit Opinion for the Increase of Regional Operational Expenditure and Own-Source Revenues Through Sensitivity Analysis in Indonesia

2020 ◽  
Vol 11 (1) ◽  
pp. 20
Author(s):  
Muhammad Ikbal Abdullah ◽  
Andi Chairil Furqan ◽  
Nina Yusnita Yamin ◽  
Fahri Eka Oktora
Keyword(s):  
Sensitivity Analysis ◽  
Significant Influence ◽  
Provincial Government ◽  
Sensitivity Testing ◽  
Audit Opinion ◽  
Absolute Value ◽  
Operating Expenditure ◽  
The Absolute ◽  
Operational Expenditure

This study aims to analyze the sensitivity testing using measurements of realization of regional own-source revenues and operating expenditure and to analyze the extent of the effect of sample differences between Java and non-Java provinces by using samples outside of Java. By using sensitivity analysis, the results found the influence of audit opinion on the performance of the provincial government mediated by the realization of regional operating expenditure. More specifically, when using the measurement of the absolute value of the realization of regional operating expenditure it was found that there was a direct positive and significant influence of audit opinion on the performance of the Provincial Government. However, no significant effect of audit opinion was found on the realization value of regional operating expenditure and the effect of the realization value of regional operating expenditure on the performance of the Provincial Government. This result implies that an increase in audit opinion will be more likely to be used as an incentive for the Provincial Government to increase the realization of regional operating expenditure.

Die Bedeutung der Homogenisationsart und -intensität für die Größe und Konstanz der Sauerstoffaufnahme von Meerschweinchen-Leberhomogenat / The Influence of Mode and Intensity of Homogenization on the Absolute Value and Stability of Oxygen Consumption of Guinea Pig Liver Homogenates

1977 ◽  
Vol 32 (11-12) ◽  
pp. 908-912 ◽  
Author(s):  
H. J. Schmidt ◽  
U. Schaum ◽  
J. P. Pichotka
Keyword(s):  
Oxygen Uptake ◽  
Guinea Pig ◽  
Measuring Time ◽  
Pig Liver ◽  
Absolute Value ◽  
Modified Method ◽  
Simultaneous Measurements ◽  
Liver Homogenates ◽  
The Absolute ◽  
Guinea Pig Liver

Abstract The influence of five different methods of homogenisation (1. The method according to Potter and Elvehjem, 2. A modification of this method called Potter S, 3. The method of Dounce, 4. Homogenisation by hypersonic waves and 5. Coarce-grained homogenisation with the “Mikro-fleischwolf”) on the absolute value and stability of oxygen uptake of guinea pig liver homogenates has been investigated in simultaneous measurements. All homogenates showed a characteristic fall of oxygen uptake during measuring time (3 hours). The modified method according to Potter and Elvehjem called Potter S showed reproducible results without any influence by homogenisation intensity.

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.

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