Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways. Transcendence The fundamental theorem of algebra tells us that if we have a non-zero polynomial with integer coefficients then that polynomial will have a root in the complex numbers. That is, for any polynomial P with integer coefficients there will be a complex number α such that P(α) = 0. Transcendence theory is concerned with the converse question, given a complex number α, is there a polynomial P with integer coefficients such that P(α) = 0? If ...

In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point. Definition Auxiliary functions are not a rigorously defined kind of function, rather they are functions which are either explicitly constructed or at least shown to exist and which provide a contradiction to some assumed hypothesis, or otherwise prove the result in question. Cr...