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Minimal Values of Polynomials for Transcendence Measure of e: Degree Two

Show simple item record 2021-10-13T13:00:03Z 2021-10-13T13:00:03Z 2021-10-13
dc.title Minimal Values of Polynomials for Transcendence Measure of e: Degree Two en
ethesis.faculty Matemaattis-luonnontieteellinen tiedekunta fi
ethesis.faculty Faculty of Science en
ethesis.faculty Matematisk-naturvetenskapliga fakulteten sv
ethesis.faculty.URI Helsingin yliopisto fi University of Helsinki en Helsingfors universitet sv
dct.creator Tiusanen, Mikko
dct.issued 2021
dct.abstract This thesis provides a program to compute minimal values of polynomials of degree two to get a transcendence measure for e, Napier’s (Neper’s) number. This is an indication of how close to zero some non-zero integer coefficient polynomial can come at e: Since e is transcendental, no such polynomial can actually attain zero. The thesis concentrates on the case of second degree polynomials. The program is written in the R5RS dialect of the programming language Scheme, a reasonably modern LISP version that offers integer and rational arithmetic only limited by the memory. The program is validated by comparison to results computed by hand, using another programming language, J. The program was also rewritten in the programming language C, providing the relevant rational number arithmetic by the library gmp. Performance characteristics of the programs are briefly compared. To appoximate the needed value of e, the programs use continued fractions. Relevant mathematical background for the subject is presented. Symmetries of the grid that represents the polynomial coefficients are used to speed up the computation: The symmetry with respect to the origin effectively halves the number of polynomial evaluations needed. The others employed are interesting but less significant. The idea came from a description of a chess endgame program of Ken Thompson by Jon Bentley, although it is likely to be older. These lead to employing the concept of a layer of polynomials with integer coefficients, the border of a punctured grid, in a sense. Layers turned out to rather nicely fit the handling of the accuracy requirements for e. The complexity of the program is still bounded from below by the number of points in the grid of polynomials, partitioned by the layers, making it bounded from below by a second degree polynomial in H, Omega(H²), where H is the natural number bounding the absolute values of the coefficients of the second and first power of e in the polynomial. The program is reasonably easily changed to handle any transcendental number other than e, in particular, if there is a convenient continued fraction to compute approximations to the number. Strictly speaking, the program does not compute the full transcendence measure, that is, a safe upper bound for each integer coefficient polynomial considered, but if this larger output is actually needed, only a minor change in the program is required. en
dct.subject Symmetry
dct.subject number theory
dct.subject transcendental number
dct.subject transcendence measure
dct.subject Napier’s (Neper’s) number e
dct.subject continued fractions
dct.subject Scheme programming language
dct.subject J programming language
dct.subject C programming language
dct.subject unlimited integer arithmetic
dct.subject unlimited rational arithmetic
dct.subject library gmp
ethesis.isPublicationLicenseAccepted true
ethesis.language englanti fi
ethesis.language English en
ethesis.language engelska sv
ethesis.thesistype pro gradu -tutkielmat fi
ethesis.thesistype master's thesis en
ethesis.thesistype pro gradu-avhandlingar sv
dct.identifier.ethesis E-thesisID:0b7185b8-6c79-4732-ba6f-0942325ec866
dct.identifier.urn URN:NBN:fi:hulib-202110133888
dc.type.dcmitype Text
dct.alternative Polynomien minimiarvoja e:n transendenssimittaa varten: aste kaksi fi
ethesis.facultystudyline Matematiikka fi
ethesis.facultystudyline Mathematics en
ethesis.facultystudyline Matematik sv
ethesis.mastersdegreeprogram Matematiikan, fysiikan ja kemian opettajan maisteriohjelma fi
ethesis.mastersdegreeprogram Master's Programme for Teachers of Mathematics, Physics and Chemistry en
ethesis.mastersdegreeprogram Magisterprogrammet för ämneslärare i matematik, fysik och kemi sv

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