EXPLICIT LOWER BOUNDS FOR LINEAR FORMS

Publication: Research - peer-reviewJournal article

Standard

EXPLICIT LOWER BOUNDS FOR LINEAR FORMS. / Leppälä, Kalle.

In: Mathematics of Computation, Vol. 85, No. 302, 11.2016, p. 2995-3008.

Publication: Research - peer-reviewJournal article

Harvard

Leppälä, K 2016, 'EXPLICIT LOWER BOUNDS FOR LINEAR FORMS' Mathematics of Computation, vol 85, no. 302, pp. 2995-3008. DOI: 10.1090/mcom/3078

APA

Leppälä, K. (2016). EXPLICIT LOWER BOUNDS FOR LINEAR FORMS. Mathematics of Computation, 85(302), 2995-3008. DOI: 10.1090/mcom/3078

CBE

Leppälä K. 2016. EXPLICIT LOWER BOUNDS FOR LINEAR FORMS. Mathematics of Computation. 85(302):2995-3008. Available from: 10.1090/mcom/3078

MLA

Leppälä, Kalle. "EXPLICIT LOWER BOUNDS FOR LINEAR FORMS". Mathematics of Computation. 2016, 85(302). 2995-3008. Available: 10.1090/mcom/3078

Vancouver

Leppälä K. EXPLICIT LOWER BOUNDS FOR LINEAR FORMS. Mathematics of Computation. 2016 Nov;85(302):2995-3008. Available from, DOI: 10.1090/mcom/3078

Author

Leppälä, Kalle / EXPLICIT LOWER BOUNDS FOR LINEAR FORMS.

In: Mathematics of Computation, Vol. 85, No. 302, 11.2016, p. 2995-3008.

Publication: Research - peer-reviewJournal article

Bibtex

@article{1b9952c26b3f465cb7351a966b8ae888,
title = "EXPLICIT LOWER BOUNDS FOR LINEAR FORMS",
keywords = "IRRATIONALITY MEASURE, LOGARITHMS, NUMBERS, PI",
author = "Kalle Leppälä",
year = "2016",
month = "11",
doi = "10.1090/mcom/3078",
volume = "85",
pages = "2995--3008",
journal = "Mathematics of Computation",
issn = "0025-5718",
publisher = "AMER MATHEMATICAL SOC",
number = "302",

}

RIS

TY - JOUR

T1 - EXPLICIT LOWER BOUNDS FOR LINEAR FORMS

AU - Leppälä,Kalle

PY - 2016/11

Y1 - 2016/11

N2 - Let I be the field of rational numbers or an imaginary quadratic field and Z(I) its ring of integers. We study some general lemmas that produce lower boundsvertical bar B-0 + B-1 theta(1) +... + B-r theta(r)vertical bar >= 1/max{vertical bar B-1 vertical bar,...,vertical bar B-r vertical bar}(mu)for all B-0,...,B-r is an element of Z(I), max{vertical bar B-1 vertical bar,...,vertical bar B-r vertical bar} >= H-0, given suitable simultaneous approximating sequences of the numbers theta(1),...,theta(r). We manage to replace the lower bound with 1/max{vertical bar B-1 vertical bar(mu 1),...,vertical bar Br vertical bar(mu r)} for all B-0,..., B-r is an element of Z(I), max{vertical bar B-1 vertical bar(mu 1),...,vertical bar Br vertical bar(mu r)} >= H-0, where the exponents mu(1),...,mu(r) are different when the given type II approximating sequences approximate some of the numbers theta(1),...,theta(r) better than the others. As an application we research certain linear forms in logarithms. Our results are completely explicit.

AB - Let I be the field of rational numbers or an imaginary quadratic field and Z(I) its ring of integers. We study some general lemmas that produce lower boundsvertical bar B-0 + B-1 theta(1) +... + B-r theta(r)vertical bar >= 1/max{vertical bar B-1 vertical bar,...,vertical bar B-r vertical bar}(mu)for all B-0,...,B-r is an element of Z(I), max{vertical bar B-1 vertical bar,...,vertical bar B-r vertical bar} >= H-0, given suitable simultaneous approximating sequences of the numbers theta(1),...,theta(r). We manage to replace the lower bound with 1/max{vertical bar B-1 vertical bar(mu 1),...,vertical bar Br vertical bar(mu r)} for all B-0,..., B-r is an element of Z(I), max{vertical bar B-1 vertical bar(mu 1),...,vertical bar Br vertical bar(mu r)} >= H-0, where the exponents mu(1),...,mu(r) are different when the given type II approximating sequences approximate some of the numbers theta(1),...,theta(r) better than the others. As an application we research certain linear forms in logarithms. Our results are completely explicit.

KW - IRRATIONALITY MEASURE

KW - LOGARITHMS

KW - NUMBERS

KW - PI

U2 - 10.1090/mcom/3078

DO - 10.1090/mcom/3078

M3 - Journal article

VL - 85

SP - 2995

EP - 3008

JO - Mathematics of Computation

T2 - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 302

ER -