I would like to tell here a story that runs over some 150 years of mathematics, around the following question: given a power series $\sum a_n T^n$ (in one variable), how can you tell it comes from a rational function?
There are two possible motivations for such a question. One comes from complex function theory: you are given an analytic function and you wish to understand its nature — the simplest of them being the rational functions, it is natural to wonder if that happens or not (the next step would be to decide whether that function is algebraic, as in the problem of Hermann Amandus Schwarz (1843–1921). Another motivation starts from the coefficients $(a_n)$, of which the power series is called the generating series; indeed, the generating series is a rational function if and only if the sequence of coefficients satisfies a linear recurrence relation.
At this stage, there are little tools to answer that question, besides is a general algebraic criterion which essentially reformulates the property that the $(a_n)$ satisfy a linear recurrence relation. For any integers $m$ and $q$, let $D_m^q$ be the determinant of size $(q+1)$ given by
\[ D_m^q = \begin{vmatrix}
a_m & a_{m+1} & \dots & a_{m+q} \\
a_{m+1} & a_{m+2} & \dots & a_{m+q+1} \\
\vdots & \vdots & & \vdots \\
a_{m+q} & a_{m+q+1} & \dots & a_{m+2q} \end{vmatrix}. \]
These determinants are called the Hankel determinants or (when $m=0$) the Kronecker determinants, from the names of the two 19th century German mathematicians Hermann Hankel (1839—1873) and Leopold von Kronecker (1823–1891). With this notation, the following properties are equivalent:
(The proof of that classic criterion is not too complicated, but the standard proof is quite smart. In his book Algebraic numbers and Fourier analysis, Raphaël Salem gives a proof which arguably easier.)
Since this algebraic criterion is very general, it is however almost impossible to prove the vanishing of these determinants without further information, and it is at this stage that Émile Borel enters the story. Émile Borel (1871–1956) has not only be a very important mathematician of the first half of the 20th century, by his works on analysis and probability theory, he also was a member of parliament, a minister of Navy, a member of Résistance during WW2. He founded the French research institution CNRS and of the Institut Henri Poincaré. He was also the first president of the Confédération des travailleurs intellectuels, a intellectual workers union.
In his 1893 paper « Sur un théorème de M. Hadamard », Borel proves the following theorem:
Theorem. — If the coefficients \(a_n\) are integers and if the power series \(\sum a_n T^n \) “defines” a function (possibly with poles) on a disk centered at the origin and of radius strictly greater than 1, then that power series is a rational function.
Observe how these two hypotheses belong to two quite unrelated worlds: the first one sets the question within number theory while the second one resorts from complex function theory. It looks almost as magic that these two hypotheses lead to the nice conclusion that the power series is a rational function.
It is also worth remarking that the second hypothesis is really necessary for the conclusion to hold, because rational functions define functions (with poles) on the whole complex plane. The status of the first hypothesis is more mysterious. While it is not necessary, the conclusion may not hold without it. For example, the exponential series \(\sum T^n/n!\) does define a function (without poles) on the whole complex plane, but is not rational (it grows too fast at infinity).
However, the interaction of number theoretical hypotheses with the question of the nature of power series was not totally inexplored at the time of Borel. For example, a 1852 theorem of the German mathematician Gotthold Eisenstein (Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Functionen) shows that when the coefficients \(a_n\) of the expansion \(\sum a_nT^n\) of an algebraic functions are rational numbers, the denominators are not arbitrary: there is an integer \(D\geq 1\) such that for all \(n\), \(a_n D^{n+1}\) is an integer. As a consequence of that theorem of Eisenstein, the exponential series or the logarithmic series cannot be algebraic.It's always time somewhere on the Internet for a mathematical proof, so that I have no excuse for avoiding to tell you *how* Émile Borel proved that result. He uses the above algebraic criterion, hence needs to prove that some determinants \(D^q_m\) introduced above do vanish (for some \(q\) and for all \(m\) large enough). Then his idea consists in observing that these determinants are integers, so that if you wish to prove that they vanish, it suffices to prove that they are smaller than one!
If non-mathematicians are still reading me, there's no mistake here: the main argument for the proof is the remark that a nonzero integer is at least one. While this may sound as a trivial remark, this is something I like to call the main theorem of number theory, because it lies at the heart of almost all proofs in number theory.
So one has to bound determinants from above, and here Borel invokes the « théorème de M. Hadamard » that a determinant, being the volume of the parallelipiped formed by the rows, is smaller than the product of the norms of these rows, considered as vectors of the Euclidean space : in 2-D, the area of a parallelogram is smaller than the lengths of its edges! (Jacques Hadamard (1865—1963) is known for many extremely important results, notably the first proof of the Prime number theorem. It is funny that this elementary result went into the title of a paper!)
But there's no hope that using Hadamard's inequality of our initial matrix can be of some help, since that matrix has integer coefficients, so that all rows have size at least one. So Borel starts making clever row combinations on the Hankel matrices that take into accounts the poles of the function that the given power series defines.
Basically, if \(f=\sum a_nT^n\), there exists a polynomial \(h=\sum c_mT^m\) such that the power series \(g=fh = \sum b_n T^n\) defines a function without poles on some disk \(D(0,R)\) where \(R>1\). Using complex function theory (Cauchy's inequalities), this implies that the coefficients \(b_n\) converge rapidly to 0, roughly as \(R^{-n}\). For the same reason, the coefficients \(a_n\) cannot grow to fast, at most as \(r^{-n}\) for some \(r>0\). The formula \(g=fh\) shows that coefficients \(b_n\) are combinations of the \(a_n\), so that the determinant \(D_n^q\) is also equal to \[ \begin{vmatrix} a_n & a_{n+1} & \dots & a_{n+q} \\ \vdots & & \vdots \\ a_{n+p-1} & a_{n+p} & \dots & a_{n+p+q-1} \\ b_{n+p} & b_{n+p+1} & & b_{n+p+q} \\ \vdots & & \vdots \\ b_{n+q} & b_{n+q+1} & \dots & b_{n+2q} \end{vmatrix}\] Now, Hadamard's inequality implies that the determinant \(D_n^q\) is (roughly) bounded above by \( (r^{-n} )^p (R^{-n}) ^{q+1-p} \): there are \(p\) lines bounded above by some \(r^{-n}\) and the next \(q+1-p\) are bounded above by \(R^{-n}\). This expression rewrites as \( 1/(r^pR^{q+1-p})^n\). Since \(R>1\), we may choose \(q\) large enough so that \(r^p R^{q+1-p}>1\), and then, when \(n\) grows to infinity, the determinant is smaller than 1. Hence it vanishes!
The next chapter of this story happens in 1928, under the hands of the Hungarian mathematician George Pólya (1887-1985). Pólya had already written several papers which explore the interaction of number theory and complex function theory, one of them will even reappear later one in this thread. In his paper “Über gewisse notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe”, he studied the analogue of Borel's question when the disk of radius \(R\) is replaced by an arbitrary domain \(U\) of the complex plane containing the origin, proving that if \(U\) is big enough, then the initial power series is a rational function. It is however not so obvious how one should measure the size of \(U\), and it is at this point that electrostatics enter the picture.
In fact, it is convenient to make an inversion : the assumption is that the series \(\sum a_n / T^n\) defines a function (with poles) on the complement of a compact subset \(K\) of the complex plane. Imagine that this compact set is made of metal, put at potential 0, and put a unit electric charge at infinity. According to the 2-D laws of electrostatics, this create an electric potential \(V_K\) which is identically \(0\) on \(K\) and behaves as \( V_K(z)\approx \log(|z|/C_K)\) at infinity. Here, \(C_K\) is a positive constant which is the capacity of \(K\).
Theorem (Pólya). — Assume that the \(a_n\) are integers and the series \(\sum a_n/T^n\) defines a function (with poles) on the complement of \(K\). If the capacity of \(K\) is \(\lt1\), then \(\sum a_n T^n\) is rational.
To apply this theorem, it is important to know of computations of capacities. This was a classic theme of complex function theory and numerical analysis some 50 years ago. Indeed, what the electric potential does is solving the Laplace equation \(\Delta V_K=0\) outside of \(K\) with Dirichlet condition on the boundary of \(K\).
In fact, the early times of complex analysis made a remarkable use of this fact. For example, it was by solving the Laplace equation that Bernhard Riemann proved the existence of meromorphic functions on “Riemann surfaces”, but analysis was not enough developed at that time (around 1860). In a stunningly creative move, Riemann imagines that his surface is partly made of metal, and partly of insulating material and he deduces the construction of the desired function from the electric potential.
More recently, complex analysis and potential theory also had applications to fluid dynamics, for example to compute (at least approximately) the flow of air outside of an airplane wing. (I am not a specialist of this, but I'd guess the development of numerical methods that run on modern computers rendered these beautiful methods obsolete.)
The relation between the theorems of Borel and Pólya is that the capacity of a disk is its radius. This can be seen by the fact that \(V(z)=\log(|z|/R\)\) solves the Laplace equation with Dirichlet condition outside of the disk of radius \(R\).
A few other capacities have been computed, not too many, in fact, because it appears to be a surprisingly difficult problem. For example, the capacity of an interval is a fourth of its length.Pólya's proof is similar to Borel's, but considers the Kronecker determinant in place of Hankel's. However, the linear combinations that will allow to show that this determinant is small are not as explicit as in Borel's proof. They follow from another interpretation of the capacity introduced by the Hungarian-Israeli mathematician Michael Fekete (1886–1957; born in then Austria–Hungary, now Serbia, he emigrated to Palestine in 1928.)
You know that the diameter \(d_2(K)\) of \(K\) is the upper bound of all distances \(|x-y|\) where \(x,y\) are arbitrary points of \(K\). Now for an integer \(n\geq 2\), consider the upper bound \(d_n(K)\) of all products of distances \( \prod_{i\neq j}{x_j-x_i}\)^{1/n(n-1)}\) where \(x_1,\dots,x_n\) are arbitrary points of \(K\). It is not so hard to prove that the sequence \(d_n(K)\) decreases with \(n\), and the limit \(\delta(K)\) of that sequence is called the transfinite diameter by Fekete.
Proposition. — \( \delta(K)= C_K\).
This allows to make a link between capacity theory and another theme of complex function theory, namely the theory of best approximation, which end up in Pólya's proof: the adequate linear combination for the \(n\)th row is given by the coefficients of the monic polynomial of degree \(n\) which has the smallest least upper bound on \(K\).
If all this is of some appeal to you, there's a wonderful little book by Thomas Ransford, Potential Theory in the Complex Plane, which I find quite readable (say, from 3rd or 4th year of math studies on).
In the forthcoming episodes, I'll discuss two striking applications of the theorems of Borel and Pólya to proof by Bernhard Dwork of a proof of a conjecture of Weil (in 1960), and by a new proof (in 1987) by Jean-Paul Bézivin and Philippe Robba of the transcendence of the numbers \(e\) and \(\pi\), two results initially proven by Charles Hermite and Ferdinand von Lindemann in 1873 and 1882.
