Difference between revisions of "The Existence of the Exponential Function"

From Drorbn
Jump to: navigation, search
(The Scheme)
Line 35: Line 35:
 
{{Equation*|<math>e_8(x+y)-e_8(x)e_8(y)=M(x,y)-(d\epsilon)(x,y)</math>.}}
 
{{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!
+
{{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}}
 
{{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>
 
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
 
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_0={\mathbb Q}[[x]]\right)\longrightarrow\left(M\in C_1={\mathbb Q}[[x,y]]\right)\longrightarrow\left(C_2={\mathbb Q}[[x,y,z]]\right)</math>,}}
 +
 +
whose first differential was already written and
  
 
==Computing the Homology==
 
==Computing the Homology==

Revision as of 20:59, 29 January 2007

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

Failed to parse (syntax error): \left\epsilon\in C_0={\mathbb Q}[[x]]\right)\longrightarrow\left(M\in C_1={\mathbb Q}[[x,y]]\right)\longrightarrow\left(C_2={\mathbb Q}[[x,y,z]]\right) ,

whose first differential was already written and

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) .