Concerning algebraic independence of some transcendental numbers

1968 ◽  
Vol 3 (1) ◽  
pp. 31-35
Author(s):  
A. A. Shmelev
Keyword(s):  
Algebraic Independence ◽  
Transcendental Numbers

Algebraic Independence Results Related to 〈q, r〉-Number Systems

2006 ◽  
Vol 147 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Shin-ichiro Okada ◽  
Iekata Shiokawa
Keyword(s):  
Algebraic Independence ◽  
Number Systems ◽  
Independence Results

ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ
Keyword(s):  
Entire Functions ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Numbers ◽  
Lecture Notes ◽  
Transcendental Entire Functions

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

scholarly journals Algebraic independence of certain numbers related to modular functions

2012 ◽  
Vol 47 (1) ◽  
pp. 121-141
Author(s):  
Carsten Elsner ◽  
Shun Shimomura ◽  
Iekata Shiokawa
Keyword(s):  
Algebraic Independence ◽  
Modular Functions

scholarly journals SMALL VALUE ESTIMATES FOR THE ADDITIVE GROUP

2010 ◽  
Vol 06 (04) ◽  
pp. 919-956 ◽  
Author(s):  
DAMIEN ROY
Keyword(s):  
Algebraic Independence ◽  
Additive Group

We generalize Gel'fond's criterion for algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of ℂ, instead of just one point (one extension deals with a subgroup of ℂ×).

scholarly journals Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable

2014 ◽  
Vol 36 (4) ◽  
pp. 1156-1166 ◽  
Author(s):  
IGORS GORBOVICKIS
Keyword(s):  
Periodic Orbits ◽  
Moduli Space ◽  
Algebraic Independence ◽  
Complex Polynomials ◽  
Generic Point ◽  
Algebraic Functions ◽  
Polynomial Maps ◽  
Affine Subspaces

We consider the space of complex polynomials of degree $n\geq 3$ with $n-1$ distinct marked periodic orbits of given periods. We prove that this space is irreducible and the multipliers of the marked periodic orbits, considered as algebraic functions on that space, are algebraically independent over $\mathbb{C}$. Equivalently, this means that at its generic point the moduli space of degree-$n$ polynomial maps can be locally parameterized by the multipliers of $n-1$ arbitrary distinct periodic orbits. We also prove a similar result for a certain class of affine subspaces of the space of complex polynomials of degree $n$.

Sign in / Sign up

Export Citation Format

Share Document