The Existence of the Exponential Function

2007-01-30T02:52:29Z

142.150.139.158: /* Computing the Homology */

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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 <math>e(x)</math> (with coefficients in <math>{\mathbb Q}</math>) which satisfies the non-linear equation

{{Equation|Main|<math>e(x+y)=e(x)e(y)</math>}}

as well as the initial condition

{{Equation|Init|<math>e(x)=1+x+</math>''(higher order terms)''.}}

Alternative proofs of the existence of <math>e(x)</math> are of course available, including the explicit formula <math>e(x)=\sum_{k=0}^\infty\frac{x^k}{k!}</math>. 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 {{EqRef|Main}} using homology. There are plenty of other examples for the use of that technique, in which the equation replacing {{EqRef|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 <math>e'=e</math>.

==The Scheme==

We aim to construct <math>e(x)</math> and solve {{EqRef|Main}} inductively, degree by degree. Equation {{EqRef|Init}} gives <math>e(x)</math> in degrees 0 and 1, and the given formula for <math>e(x)</math> indeed solves {{EqRef|Main}} in degrees 0 and 1. So booting the induction is no problem. Now assume we've found a degree 7 polynomial <math>e_7(x)</math> which solves {{EqRef|Main}} up to and including degree 7, but at this stage of the construction, it may well fail to solve {{EqRef|Main}} in degree 8. Thus modulo degrees 9 and up, we have

{{Equation|M|<math>e_7(x+y)-e_7(x)e_7(y)=M(x,y)</math>,}}

where <math>M(x,y)</math> is the "mistake for <math>e_7</math>", a certain homogeneous polynomial of degree 8 in the variables <math>x</math> and <math>y</math>.

Our hope is to "fix" the mistake <math>M</math> by replacing <math>e_7(x)</math> with <math>e_8(x)=e_7(x)+\epsilon(x)</math>, where <math>\epsilon_8(x)</math> is a degree 8 "correction", a homogeneous polynomial of degree 8 in <math>x</math> (well, in this simple case, just a multiple of <math>x^8</math>).

{{Begin Side Note|35%}}*1 The terms containing no <math>\epsilon</math>'s make a copy of the left hand side of {{EqRef|M}}. The terms linear in <math>\epsilon</math> are <math>\epsilon(x+y)</math>, <math>-e_7(x)\epsilon(y)</math> and <math>-\epsilon(x)e_7(y)</math>. Note that since the constant term of <math>e_7</math> is 1 and since we only care about degree 8, the last two terms can be replaced by <math>-\epsilon(y)</math> and <math>-\epsilon(x)</math>, respectively. Finally, we don't even need to look at terms higher than linear in <math>\epsilon</math>, for these have degree 16 or more, high in the stratosphere.
{{End Side Note}}
So we substitute <math>e_8(x)=e_7(x)+\epsilon(x)</math> into <math>e(x+y)-e(x)e(y)</math> (a version of {{EqRef|Main}}), expand, and consider only the low degree terms - those below and including degree 8:<sup>*1</sup>

{{Equation*|<math>e_8(x+y)-e_8(x)e_8(y)=M(x,y)-\epsilon(y)+\epsilon(x+y)-\epsilon(x)</math>.}}

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

{{Equation*|<math>e_8(x+y)-e_8(x)e_8(y)=M(x,y)-(d\epsilon)(x,y)</math>.}}

{{Begin Side Note|35%}}*2 It is worth noting that in some a priori sense the existence of an exponential function, a solution of <math>e(x+y)=e(x)e(y)</math>, is quite unlikely. For <math>e</math> must be an element of the relatively small space <math> {\mathbb Q}[[x]] </math> of power series in one variable, but the equation it is required to satisfy lives in the much bigger space <math> {\mathbb Q}[[x,y]] </math>. Thus in some sense we have more equations than unknowns and a solution is unlikely. How fortunate we are!
{{End Side Note}}
To continue with our inductive construction we need to have that <math>e_8(x+y)-e_8(x)e_8(y)=0</math>. Hence the existence of the exponential function hinges upon our ability to find an <math>\epsilon</math> for which <math>M=d\epsilon</math>. In other words, we must show that <math>M</math> is in the image of <math>d</math>. This appears hopeless unless we learn more about <math>M</math>, for the domain space of <math>d</math> is much smaller than its target space and thus <math>d</math> cannot be surjective, and if <math>M</math> was in any sense "random", we simply wouldn't be able to find our correction term <math>\epsilon</math>.<sup>*2</sup>

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

{{Equation*|<math>\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)</math>,}}

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

==Computing the Homology==

So what kind of relations can we get for <math>M</math>? Well, it measures how close <math>e_7</math> 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 <math>M(x,y)=M(y,x)</math>, and indeed this is obvious from the definition. Now let's try associativity, that is, let's compute <math>e_7(x+y+z)</math> associating first as <math>(x+y)+z</math> and then as <math>x+(y+z)</math>. In the first way we get

<math>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).</math>

In the second we get

<math>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)</math>.

Comparing these two we get an interesting relation for <math>M</math>: <math>M(x+y,z)+M(x,y)e_7(z) = M(x,y+z) + e_7(x)M(y,z) </math>.

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The Existence of the Exponential Function