Anne-Maria Ernvall-Hytönen

On the transcendence measure of e (joint work with Tapani Matala-aho and Louna Seppälä)

A number is called algebraic if it is a root of a polynomial with integer coefficients. If it is not algebraic, then it it transcendental. The constant e is a famous example of a transcendental number. Therefore, it makes sense to study how close to zero can we get if we evaluate polynomials with integers coefficients at e. Clearly, this bound depends on the height and degree of the polynomial. Suitably normalized bounds of this type are called transcendence measures. The question about the correct size of the transcendence measure of e is an old one. The work started with Borel in 1899, and has been continued by several people, including Popken and Mahler. Hata's bound in 1995 was the best known until very recently, we were able to improve this. During my talk, I will explain the critical ingredients in the proof and what was needed to improve Hata's result. Finally, I will briefly describe some generalizations.