The Existence of the Exponential Function

Paperlets Navigation Panel   Algebraic Knot Theory Algebraic Knot Theory - A Call for Action. The Kontsevich Integral for Knotted Trivalent Graphs. Knotted Trivalent Graphs are Finitely Presented. VS, TS and TG Algebras. The Alexander Polynomial of a Knotted Trivalent Graph. Some Questions About Trinions. Finite Type Invariants The Envelope of a Finite Type Invariant. The Envelope of The Alexander Polynomial. Some Quotients of A. Associators with Frozen Feet. The HOMFLY Braidor Algebra. Coloured Chord Diagrams. Khovanov Homology Mutation Invariance of Khovanov Homology. Khovanov Homology of Alternating Tangles. Other The Existence of the Exponential Function. The Many Faces of Alexander.

Introduction

The purpose of this paperlet is to use some homological algebra in order to prove the existence of a power series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e(x)} (with coefficients in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Q}} ) which satisfies the non-linear equation

[Main]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} for which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=d\epsilon} . In other words, we must show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is in the image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} . This appears hopeless unless we learn more about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , for the domain space of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is much smaller than its target space and thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} cannot be surjective, and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} was in any sense "random", we simply wouldn't be able to find our correction term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} .^*2

As we shall see momentarily by "finding syzygies", $\epsilon$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} fit within the 0th and 1st chain groups of a rather short complex

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (d^2m)(x,y,z)=m(y,z)-m(x+y,z)+m(x,y+z)-m(x,y)} for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\in{\mathbb Q}[[x,y]]} . We shall further see that for "our" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , we have $d^{2}M=0$. Therefore in order to show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is in the image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^1} , it suffices to show that the kernel of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^2} is equal to the image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^1} , or simply that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^2=0} .

Finding a Syzygy

So what kind of relations can we get for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} ? Well, it measures how close Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(x,y)=M(y,x)} , and indeed this is obvious from the definition. Now let's try associativity, that is, let's compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_7(x+y+z)} associating first as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x+y)+z} and then as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+(y+z)} . In the first way we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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).}

In the second we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)} .

Comparing these two we get an interesting relation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(x+y,z)+M(x,y)e_7(z) = M(x,y+z) + e_7(x)M(y,z) } . Since we'll only use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} to find the next highest term, we can be sloppy about all but the first term of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . This means that in the relation we just found we can replace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_7} by its constant term, namely 1. Upon rearranging, we get the relation promised for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^2M = M(y,z)-M(x+y,z)+M(x,y+z)-M(x,y) = 0} .

Computing the Homology

Now let's prove that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^2=0} for our (piece of) chain complex. That is, letting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(x,y) \in \mathbb{Q}[[x,y]]} be such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^2M=0} , we'll prove that for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(x) \in \mathbb{Q}[[x]]} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^1E = M } .

Write the two power series as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(x) = \sum{\frac{e_i}{i!}x^i} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(x,y) = \sum{\frac{m_{ij}}{i!j!}x^i y^j} } , where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_i} are the unknowns we wish to solve for.

The coefficient of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^i y^j z^k} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(y,z)-M(x+y,z)+M(x,y+z)-M(x,y)} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta_{i0} m_{jk}}{j!k!} - \binom{i+j}{i} \frac{m_{i+j,k}}{(i+j)!k!} + \binom{j+k}{j} \frac{m_{i,j+k}}{i!(j+k)!} - \frac{\delta_{k0} m_{ij}}{i!j!}. }

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{i0} } is a Kronecker delta: 1 if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=0 } and 0 otherwise. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^2 M = 0 } , this coefficient should be zero. Multiplying by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i!j!k! } (and noting that, for example, the first term doesn't need an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i! } since the delta is only nonzero when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=0 } ) we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{i0} m_{jk} - m_{i+j,k} + m_{i,j+k} + \delta_{k0} m_{ij}} .

An entirely analogous procedure tells us that the equations we must solve boil down to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_[i0} e_j - e_{i+j} + \delta_{j0} e_i = m_{ij} } .