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_8,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).

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. (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!)

Computing the Homology