Hi Bo, I see you have a good fluency in formal power-series and I'd like to know if you have any immediate insight about the possibility/problems of two-variables power-series satisfying a "formal" Ackermann/Goodstein equation.
Basically formal power-series over a ring \(R[[X]]\) are formally the same as functions \(R^\mathbb N\) but with much richer algebraic structure on it. Formal power-series on two variables are something like infinite matrices \(R[[X,Y]]\simeq R^{\mathbb N\times \mathbb N} \).
In some cases maybe it is possible to nest-compose 2-variables formal power-series. What about a powerserie \(A\in \mathbb R[[X,Y]]\) s.t.
\[A(S_0(X),S_1(Y))=A(X,A(S_0(X),Y))\]
Where \(S_0(X)=1+X\) and \(S_1(Y)=1+Y\).
What would be the condition to impose on the coefficient matrix of \(A\) that ensure the existence of that composition "as a formal power-series", and what that equation would imply on the coefficient matrix itself?
Inspired by the discussion in (wikipedia) Formal group law.
Let \(A\in R[[X,Y]] \) and \(A(X,Y)=\sum_{n,m}a_{n,m}X^nY^m\). If I'm not mistaken, the condition \(A(0,Y)=Y+1\) implies that \(a_{0,0}=1\), \(a_{0,1}=1\) and if \(1\lt m\) we have \(a_{0,m}=0\).
\[
A=
\begin{bmatrix}
1 & 1 & 0 & 0 & \cdots \\
a_{10} & a_{11} & a_{12} & a_{13} &\cdots \\
a_{20} & a_{21} & a_{22} & a_{23} &\cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots \\
\end{bmatrix}\]
So the initial condition, what I call "trivial zeration" or Goodstein condition, implies that
\[A(X,Y)=1+Y+\sum_{0\lt n,m}a_{n,m}X^nY^n\]
Basically formal power-series over a ring \(R[[X]]\) are formally the same as functions \(R^\mathbb N\) but with much richer algebraic structure on it. Formal power-series on two variables are something like infinite matrices \(R[[X,Y]]\simeq R^{\mathbb N\times \mathbb N} \).
In some cases maybe it is possible to nest-compose 2-variables formal power-series. What about a powerserie \(A\in \mathbb R[[X,Y]]\) s.t.
\[A(S_0(X),S_1(Y))=A(X,A(S_0(X),Y))\]
Where \(S_0(X)=1+X\) and \(S_1(Y)=1+Y\).
What would be the condition to impose on the coefficient matrix of \(A\) that ensure the existence of that composition "as a formal power-series", and what that equation would imply on the coefficient matrix itself?
Inspired by the discussion in (wikipedia) Formal group law.
Let \(A\in R[[X,Y]] \) and \(A(X,Y)=\sum_{n,m}a_{n,m}X^nY^m\). If I'm not mistaken, the condition \(A(0,Y)=Y+1\) implies that \(a_{0,0}=1\), \(a_{0,1}=1\) and if \(1\lt m\) we have \(a_{0,m}=0\).
\[
A=
\begin{bmatrix}
1 & 1 & 0 & 0 & \cdots \\
a_{10} & a_{11} & a_{12} & a_{13} &\cdots \\
a_{20} & a_{21} & a_{22} & a_{23} &\cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots \\
\end{bmatrix}\]
So the initial condition, what I call "trivial zeration" or Goodstein condition, implies that
\[A(X,Y)=1+Y+\sum_{0\lt n,m}a_{n,m}X^nY^n\]
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
