The Existence of the Exponential Function

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Introduction

The purpose of this paperlet is to use some homological algebra in order to prove the existence of a power series $e(x)$ (with coefficients in ${\mathbb {Q} }$) which satisfies the non-linear equation

[Main]

$e(x+y)=e(x)e(y)$

as well as the initial condition

[Init]

$e(x)=1+x+$(higher order terms).

Alternative proofs of the existence of $e(x)$ are of course available, including the explicit formula $e(x)=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}$. Thus the value of this paperlet is not in the result it proves but rather in the allegorical story it tells: that there is a technique to solve functional equations such as [Main] using homology. There are plenty of other examples for the use of that technique, in which the equation replacing [Main] isn't as easy. Thus the exponential function seems to be the easiest illustration of a general principle and as such it is worthy of documenting.

Thus below we will pretend not to know the exponential function and/or its relationship with the differential equation $e'=e$.

The Scheme

We aim to construct $e(x)$ and solve [Main] inductively, degree by degree. Equation [Init] gives $e(x)$ in degrees 0 and 1, and the given formula for $e(x)$ indeed solves [Main] in degrees 0 and 1. So booting the induction is no problem. Now assume we've found a degree 7 polynomial $e_{7}(x)$ which solves [Main] up to and including degree 7, but at this stage of the construction, it may well fail to solve [Main] in degree 8. Thus modulo degrees 9 and up, we have

[M]

$e_{7}(x+y)-e_{7}(x)e_{7}(y)=M(x,y)$,

where $M(x,y)$ is the "mistake for $e_{7}$", a certain homogeneous polynomial of degree 8 in the variables $x$ and $y$.

Our hope is to "fix" the mistake $M$ by replacing $e_{7}(x)$ with $e_{8}(x)=e_{7}(x)+\epsilon (x)$, where $\epsilon _{8}(x)$ is a degree 8 "correction", a homogeneous polynomial of degree 8 in $x$ (well, in this simple case, just a multiple of $x^{8}$).

*1 The terms containing no $\epsilon$'s make a copy of the left hand side of [M]. The terms linear in $\epsilon$ are $\epsilon (x+y)$, $-e_{7}(x)\epsilon (y)$ and $-\epsilon (x)e_{7}(y)$. Note that since the constant term of $e_{7}$ is 1 and since we only care about degree 8, the last two terms can be replaced by $-\epsilon (y)$ and $-\epsilon (x)$, respectively. Finally, we don't even need to look at terms higher than linear in $\epsilon$, for these have degree 16 or more, high in the stratosphere.

So we substitute $e_{8}(x)=e_{7}(x)+\epsilon (x)$ into $e(x+y)-e(x)e(y)$ (a version of [Main]), expand, and consider only the low degree terms - those below and including degree 8:^*1

$e_{8}(x+y)-e_{8}(x)e_{8}(y)=M(x,y)-\epsilon (y)+\epsilon (x+y)-\epsilon (x)$.

We define a "differential" $d:{\mathbb {Q} }[x]\to {\mathbb {Q} }[x,y]$ by $(df)(x,y)=f(y)-f(x+y)+f(x)$, and the above equation becomes

$e_{8}(x+y)-e_{8}(x)e_{8}(y)=M(x,y)-(d\epsilon )(x,y)$.

*2 It is worth noting that in some a priori sense the existence of an exponential function, a solution of $e(x+y)=e(x)e(y)$, is quite unlikely. For $e$ must be an element of the relatively small space ${\mathbb {Q} }[[x]]$ of power series in one variable, but the equation it is required to satisfy lives in the much bigger space ${\mathbb {Q} }[[x,y]]$. Thus in some sense we have more equations than unknowns and a solution is unlikely. How fortunate we are!

To continue with our inductive construction we need to have that $e_{8}(x+y)-e_{8}(x)e_{8}(y)=0$. Hence the existence of the exponential function hinges upon our ability to find an $\epsilon$ for which $M=d\epsilon$. In other words, we must show that $M$ is in the image of $d$. This appears hopeless unless we learn more about $M$, for the domain space of $d$ is much smaller than its target space and thus $d$ cannot be surjective, and if $M$ was in any sense "random", we simply wouldn't be able to find our correction term $\epsilon$.^*2

As we shall see momentarily by "finding syzygies", $\epsilon$ and $M$ fit within the 0th and 1st chain groups of a rather short complex

$\left(\epsilon \in C_{1}={\mathbb {Q} }[[x]]\right)\longrightarrow \left(M\in C_{2}={\mathbb {Q} }[[x,y]]\right)\longrightarrow \left(C_{3}={\mathbb {Q} }[[x,y,z]]\right)$,

whose first differential was already written and whose second differential is given by $(d^{2}m)(x,y,z)=m(y,z)-m(x+y,z)+m(x,y+z)-m(x,y)$ for any $m\in {\mathbb {Q} }[[x,y]]$. We shall further see that for "our" $M$, we have $d^{2}M=0$. Therefore in order to show that $M$ is in the image of $d^{1}$, it suffices to show that the kernel of $d^{2}$ is equal to the image of $d^{1}$, or simply that $H^{2}=0$.

Computing the Homology

So what kind of relations can we get for $M$? Well, it measures how close $e_{7}$ is to turning sums into products, so we can look for preservation of properties that both addition and multiplication have. For example, they're both commutative, so we should have $M(x,y)=M(y,x)$, and indeed this is obvious from the definition. Now let's try associativity, that is, let's compute $e_{7}(x+y+z)$ associating first as $(x+y)+z$ and then as $x+(y+z)$. In the first way we get

[$e_{7}(x+y+z)=M(x+y,z)+e_{7}(x+y)e_{7}(z)=M(x+y,z)+\left(M(x,y)+e_{7}(x)e_{7}(y)\right)e_{7}(z)$.]

{{{2}}}

In the second we get

[$e_{7}(x+y+z)=M(x,y+z)+e_{7}(x)e_{7}(y+z)=M(x+y,z)+e_{7}(x)\left(M(y,z)+e_{7}(y)e_{7}(z)\right)$.]

{{{2}}}

Comparing these two we get an interesting relation for $M$: $M(x+y,z)+M(x,y)e_{7}(z)=M(x,y+z)+e_{7}(x)M(y,z)$.