Liouville's transcendence criterion

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. One of these first results was Liouville's proof that transcendental numbers exist when he showed that the so called Liouville numbers were transcendental.^[2] He did this by discovering a transcendence criterion which these numbers satisfied. To derive this criterion he started with a general algebraic number α and found some property that this number would necessarily satisfy. The auxiliary function he used in the course of proving this criterion was simply the minimal polynomial of α, which is the irreducible polynomial f with integer coefficients such that f(α) = 0. This function can be used to estimate how well the algebraic number α can be estimated by rational numbers p/q. Specifically if α has degree d at least two then he showed that

${\displaystyle \left|f\left({\frac {p}{q}}\right)\right|\geq {\frac {1}{q^{d}}},}$

and also, using the mean value theorem, that there is some constant depending on α, say c(α), such that

${\displaystyle \left|f\left({\frac {p}{q}}\right)\right|\leq c(\alpha )\left|\alpha -{\frac {p}{q}}\right|.}$

Combining these results gives a property that the algebraic number must satisfy; therefore any number not satisfying this criterion must be transcendental.

The auxiliary function in Liouville's work is very simple, merely a polynomial that vanishes at a given algebraic number. This kind of property is usually the one that auxiliary functions satisfy. They either vanish or become very small at particular points, which is usually combined with the assumption that they do not vanish or can't be too small to derive a result.

Fourier's proof of the irrationality of e

Another simple, early occurrence is in Fourier's proof of the irrationality of e,^[3] though the notation used usually disguises this fact. Fourier's proof used the power series of the exponential function:

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}$

By truncating this power series after, say, N + 1 terms we get a polynomial with rational coefficients of degree N which is in some sense "close" to the function e^x. Specifically if we look at the auxiliary function defined by the remainder:

${\displaystyle R(x)=e^{x}-\sum _{n=0}^{N}{\frac {x^{n}}{n!}}}$

then this function—an exponential polynomial—should take small values for x close to zero. If e is a rational number then by letting x = 1 in the above formula we see that R(1) is also a rational number. However, Fourier proved that R(1) could not be rational by eliminating every possible denominator. Thus e cannot be rational.

Hermite's proof of the irrationality of e^r

Hermite extended the work of Fourier by approximating the function e^x not with a polynomial but with a rational function, that is a quotient of two polynomials. In particular he chose polynomials A(x) and B(x) such that the auxiliary function R defined by

${\displaystyle R(x)=B(x)e^{x}-A(x)}$

could be made as small as he wanted around x = 0. But if e^r were rational then R(r) would have to be rational with a particular denominator, yet Hermite could make R(r) too small to have such a denominator, hence a contradiction.

Hermite's proof of the transcendence of e

To prove that e was in fact transcendental, Hermite took his work one step further by approximating not just the function e^x, but also the functions e^kx for integers k = 1,...,m, where he assumed e was algebraic with degree m. By approximating e^kx by rational functions with integer coefficients and with the same denominator, say A_k(x) / B(x), he could define auxiliary functions R_k(x) by

${\displaystyle R_{k}(x)=B(x)e^{kx}-A_{k}(x).}$

For his contradiction Hermite supposed that e satisfied the polynomial equation with integer coefficients a₀ + a₁e + ... + a_me^m = 0. Multiplying this expression through by B(1) he noticed that it implied

${\displaystyle R=a_{0}+a_{1}R_{1}(1)+\cdots +a_{m}R_{m}(1)=a_{1}A_{1}(1)+\cdots +a_{m}A_{m}(1).}$

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.

Auxiliary polynomial theorem

Let β equal the cube root of b/a in the equation ax³ + bx³ = c and assume m is an integer that satisfies m + 1 > 2n/3 ≥ m ≥ 3 where n is a positive integer.

Then there exists

${\displaystyle F(X,Y)=P(X)+Y*Q(X)}$

such that

${\displaystyle \sum _{i=0}^{m+n}u_{i}X^{i}=P(X),}$

${\displaystyle \sum _{i=0}^{m+n}v_{i}X^{i}=Q(X).}$

The auxiliary polynomial theorem states

${\displaystyle \max _{0\leq i\leq m+n}{(|u_{i}|,|v_{i}|)}\leq 2b^{9(m+n)}.}$

A theorem of Lang

In the 1960s Serge Lang proved a result using this non-explicit form of auxiliary functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems.^[8] The theorem deals with a number field K and meromorphic functions f₁,...,f_N of order at most ρ, at least two of which are algebraically independent, and such that if we differentiate any of these functions then the result is a polynomial in all of the functions. Under these hypotheses the theorem states that if there are m distinct complex numbers ω₁,...,ω_m such that f_i (ω_j ) is in K for all combinations of i and j, then m is bounded by

${\displaystyle m\leq 20\rho [K:\mathbb {Q} ].}$

To prove the result Lang took two algebraically independent functions from f₁,...,f_N, say f and g, and then created an auxiliary function which was simply a polynomial F in f and g. This auxiliary function could not be explicitly stated since f and g are not explicitly known. 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,...,ω_m. Because of this high order vanishing it can be shown that a high-order derivative of F takes a value of small size one of the ω_is, "size" here referring to an algebraic property of a number. 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.

A proof of the Hermite–Lindemann theorem

One of the simpler applications of this method is a proof of the real version of the Hermite–Lindemann theorem. That is, if α is a non-zero, real algebraic number, then e^α is transcendental. First we let k be some natural number and n be a large multiple of k. The interpolation determinant considered is the determinant Δ of the n⁴×n⁴ matrix

${\displaystyle \left(\{\exp(j_{2}x)x^{j_{1}-1}\}^{(i_{1}-1)}{\Big |}_{x=(i_{2}-1)\alpha }\right).}$

The rows of this matrix are indexed by 1 ≤ i₁ ≤ n⁴/k and 1 ≤ i₂ ≤ k, while the columns are indexed by 1 ≤ j₁ ≤ n³ and 1 ≤ j₂ ≤ n. So the functions in our matrix are monomials in x and e^x and their derivatives, and we are interpolating at the k points 0,α,2α,...,(k − 1)α. Assuming that e^α is algebraic we can form the number field Q(α,e^α) of degree m over Q, and then multiply Δ by a suitable denominator as well as all its images under the embeddings of the field Q(α,e^α) into C. For algebraic reasons this product is necessarily an integer, and using arguments relating to Wronskians it can be shown that it is non-zero, so its absolute value is an integer Ω ≥ 1.

Using a version of the mean value theorem for matrices it is possible to get an analytic bound on Ω as well, and in fact using big-O notation we have

${\displaystyle \Omega =O\left(\exp \left(\left({\frac {m+1}{k}}-{\frac {3}{2}}\right)n^{8}\log n\right)\right).}$

The number m is fixed by the degree of the field Q(α,e^α), but k is the number of points we are interpolating at, and so we can increase it at will. And once k > 2(m + 1)/3 we will have Ω → 0, eventually contradicting the established condition Ω ≥ 1. Thus e^α cannot be algebraic after all.^[11]