Because of the naming convention mentioned above, auxiliary functions can be dated back to their source simply by looking at the earliest results in transcendence theory.
The auxiliary function in Liouville's work is very simple, merely a polynomial that vanishes at a given algebraic number.
Specifically if we look at the auxiliary function defined by the remainder: then this function—an exponential polynomial—should take small values for x close to zero.
Multiplying this expression through by B(1) he noticed that it implied The right hand side is an integer and so, by estimating the auxiliary functions and proving that 0 < |R| < 1 he derived the necessary contradiction.
A breakthrough by Axel Thue and Carl Ludwig Siegel in the twentieth century was the realisation that these functions don't necessarily need to be explicitly known – it can be enough to know they exist and have certain properties.
[5] This method was picked up on and used by several other mathematicians, including Alexander Gelfond and Theodor Schneider who used it independently to prove the Gelfond–Schneider theorem.
But using Siegel's lemma Lang showed how to make F in such a way that it vanished to a high order at the m complex numbers ω1,...,ωm.
Using the maximum modulus principle Lang also found a separate way to estimate the absolute values of derivatives of F, and using standard results comparing the size of a number and its absolute value he showed that these estimates were contradicted unless the claimed bound on m holds.
After the myriad of successes gleaned from using existent but not explicit auxiliary functions, in the 1990s Michel Laurent introduced the idea of interpolation determinants.
Since a determinant is just a polynomial in the entries of a matrix, these auxiliary functions succumb to study by analytic means.
A development by Jean-Benoît Bost removed this problem with the use of Arakelov theory,[10] and research in this area is ongoing.