Non-Archimedean Analysis on the Extended Hyperreal Line *Rd and the Solution of Some Very Old Transcendence Conjectures over the Field Q
Author(s)
Jaykov Foukzon*
ABSTRACT
In 1980 F. Wattenberg constructed the Dedekind completion *Rd of the Robinson non-archimedean field *R and established basic algebraic properties of *Rd. In 1985 H. Gonshor established further fundamental properties of *Rd. In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion *Rd in transcendental number theory were considered. Given any analytic function of one complex variable
, we investigate the arithmetic nature of the values of 
at transcendental points 
. Main results are: 1) the both numbers 
and 
are irrational; 2) number ee is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.
In 1980 F. Wattenberg constructed the Dedekind completion *Rd of the Robinson non-archimedean field *R and established basic algebraic properties of *Rd. In 1985 H. Gonshor established further fundamental properties of *Rd. In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion *Rd in transcendental number theory were considered. Given any analytic function of one complex variable
, we investigate the arithmetic nature of the values of at transcendental points . Main results are: 1) the both numbers and are irrational; 2) number ee is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.
KEYWORDS
Non-Archimedean Analysis, Robinson Non-Archimedian Field, Dedekind Completion, Dedekind Hyperreals, Wattenberg Embeding, Gonshor Idempotent Theory, Gonshor Transfer
Non-Archimedean Analysis, Robinson Non-Archimedian Field, Dedekind Completion, Dedekind Hyperreals, Wattenberg Embeding, Gonshor Idempotent Theory, Gonshor Transfer
Cite this paper
Foukzon, J. (2015) Non-Archimedean Analysis on the Extended Hyperreal Line *Rd and the Solution of Some Very Old Transcendence Conjectures over the Field Q. Advances in Pure Mathematics, 5, 587-628. doi: 10.4236/apm.2015.510056.
Foukzon, J. (2015) Non-Archimedean Analysis on the Extended Hyperreal Line *Rd and the Solution of Some Very Old Transcendence Conjectures over the Field Q. Advances in Pure Mathematics, 5, 587-628. doi: 10.4236/apm.2015.510056.
References
[1] Nesterenko, Y.V. and Philippon, P., Eds. (2001) Introduction to Algebraic Independence Theory. Series: Lecture Notes in Mathematics, Vol. 1752. XIII, Springer Science & Business Media, 256 pp. [2] Waldschmidt, M. (2003) Algebraic Values of Analytic Functions. Journal of Computational and Applied Mathematics, 160, 323-333.
http://dx.doi.org/10.1016/S0377-0427(03)00637-X [3] Foukzon, J. (2006) The Solution of one Very Old Problem in Transcendental Numbers Theory. Spring Central Sectional Meeting Notre Dame, IN, 8-9 April 2006, Meeting #1016 Preliminary Report.
http://www.ams.org/meetings/sectional/1016-11-8.pdf [4] Foukzon, J. (2013) Non-Archimedean Analysis on the Extended Hyperreal Line *Rd and Some Transcendence Conjectures over Field Q and *Qω.
http://arxiv.org/abs/0907.0467 [5] Goldblatt, R. (1998) Lectures on the Hyperreals. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-0615-6 [6] Wattenberg, F. (1980) [0, ∞]-Valued, Translation Invariant Measures on and the Dedekind Completion of . Pacific Journal of Mathematics, 90, 223-247.
http://dx.doi.org/10.2140/pjm.1980.90.223 [7] Gonshor, H. (1985) Remarks on the Dedekind Completion of a Nonstandard Model of the Reals. Pacific Journal of Mathematics, 118, 117-132.
http://dx.doi.org/10.2140/pjm.1985.118.117 [8] Shidlovsky, A.B. (1982) Diophantine Approximations and Transcendental Numbers. Moscow State University, Moscov. (In Russian).
http://en.bookfi.org/book/506517
http://bookre.org/reader?file=506517&pg=129
[1] Nesterenko, Y.V. and Philippon, P., Eds. (2001) Introduction to Algebraic Independence Theory. Series: Lecture Notes in Mathematics, Vol. 1752. XIII, Springer Science & Business Media, 256 pp. [2] Waldschmidt, M. (2003) Algebraic Values of Analytic Functions. Journal of Computational and Applied Mathematics, 160, 323-333.
http://dx.doi.org/10.1016/S0377-0427(03)00637-X [3] Foukzon, J. (2006) The Solution of one Very Old Problem in Transcendental Numbers Theory. Spring Central Sectional Meeting Notre Dame, IN, 8-9 April 2006, Meeting #1016 Preliminary Report.
http://www.ams.org/meetings/sectional/1016-11-8.pdf [4] Foukzon, J. (2013) Non-Archimedean Analysis on the Extended Hyperreal Line *Rd and Some Transcendence Conjectures over Field Q and *Qω.
http://arxiv.org/abs/0907.0467 [5] Goldblatt, R. (1998) Lectures on the Hyperreals. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-0615-6 [6] Wattenberg, F. (1980) [0, ∞]-Valued, Translation Invariant Measures on and the Dedekind Completion of . Pacific Journal of Mathematics, 90, 223-247.
http://dx.doi.org/10.2140/pjm.1980.90.223 [7] Gonshor, H. (1985) Remarks on the Dedekind Completion of a Nonstandard Model of the Reals. Pacific Journal of Mathematics, 118, 117-132.
http://dx.doi.org/10.2140/pjm.1985.118.117 [8] Shidlovsky, A.B. (1982) Diophantine Approximations and Transcendental Numbers. Moscow State University, Moscov. (In Russian).
http://en.bookfi.org/book/506517
http://bookre.org/reader?file=506517&pg=129