3 added 2 characters in body

The addition of $p$-typical Witt vectors ($p$ a prime number) is given by universal polynomials $S_n=S_n(X_0,\dots,X_n;Y_0,\dots,Y_n)\in\mathbb{Z}[X_0,X_1,\dots;Y_0,Y_1,\dots]$ determined by the equalities

$\Phi_n(S_0,\dots,S_n)=\Phi_n(X_0,\dots,X_n)+\Phi_n(Y_0,\dots,Y_n)$ for all $n\ge 0$,

where

$\Phi(T_0,\dots,T_n)=(T_0)^{p^n}+p(T_1)^{p^{n-1}}+\dots+p^nT_n$\Phi_n(T_0,\dots,T_n)=(T_0)^{p^n}+p(T_1)^{p^{n-1}}+\dots+p^nT_n$. I guess that whoever sees the Witt vectors for the first time writes down explicitly $S_0=X_0+Y_0$, $S_1=X_1+Y_1+\frac{1}{p}((X_0)^p+(Y_0)^p-(X_0+Y_0)^p)$, maybe $S_2$ if she/he is courageous, and then stops since the computation becomes extremely messy. I think that there is no reasonable explicit expression in general, but patterns seem to exist and my question is about making these patterns more precise. Before I ask, let me illustrate with $S_2$. It is easy to see that there exists a unique sequence of polynomials $R_n\in\mathbb{Z}[X,Y]$, $n\ge 0$, such that $X^{p^n}+Y^{p^n}=R_0(X,Y)^{p^n}+pR_1(X,Y)^{p^{n-1}}+\dots+p^nR_n(X,Y)$. For example $R_0=X+Y$ and $R_1=\frac{1}{p}(X^p+Y^p-(X+Y)^p)$. Then: $S_1=R_0(X_1,Y_1)+R_1(X_0,Y_0)$ $S_2=R_0(X_2,Y_2)+R_1(X_1,Y_1)+R_1(R_0(X_1,Y_1),R_1(X_0,Y_0))+R_2(X_0,Y_0)$. Can someone make the shape of $S_n$ more precise, e.g. in the form $S_n=P_0+\dots+P_n$ presumably with $P_0=R_0(X_n,Y_n)$, $P_n=R_n(X_0,Y_0)$? The intermediary $P_i$'s are more complicated but should be (uniquely) determined by a condition of the type "$P_i$ is an iterated composition of $R_i$ involving only the variables $X_0,\dots,X_i$". Maybe the polynomials $P_i$ should be homogeneous w.r.t. some graduation. Any hint or relevant reference will be appreciated. Thanks! 2 added 5 characters in body The addition of $p$-typical Witt vectors ($p$ a prime number) is given by universal polynomials $S_n=S_n(X_0,\dots,X_n;Y_0,\dots,Y_n)\in\mathbb{Z}[X_0,X_1,\dots;Y_0,Y_1,\dots]$ determined by the equalities $\Phi_n(S_0,\dots,S_n)=\Phi_n(X_0,\dots,X_n)+\Phi_n(Y_0,\dots,Y_n)$ for all $n\ge 0$, where $\Phi(T_0,\dots,T_n)=(T_0)^{p^n}+p(T_1)^{p^{n-1}}+\dots+p^nT_n$. I guess that whoever sees the Witt vectors for the first time writes down explicitly $S_0=X_0+Y_0$, $S_1=X_1+Y_1+\frac{1}{p}((X_0)^p+(Y_0)^p-(X_0+Y_0)^p)$, maybe $S_2$ if she/he is courageous, and then stops since the computation becomes extremely messy. I think that there is no reasonable explicit expression in general, but patterns seem to exist and my question is about making these patterns more precise. Before I ask, let me illustrate with $S_2$. It is easy to see that there exists a unique sequence of polynomials $R_n\in\mathbb{Z}[X,Y]$, $n\ge 0$, such that $X^{p^n}+Y^{p^n}=R_0(X,Y)^{p^n}+pR_1(X,Y)^{p^{n-1}}+\dots+p^nR_n(X,Y)$. For example $R_0=X+Y$ and $R_1=\frac{1}{p}(X^p+Y^p-(X+Y)^p)$. Then: $S_1=R_0(X_1,Y_1)+R_1(X_0,Y_0)$ $S_2=R_0(X_2,Y_2)+R_1(X_1,Y_1)+R_1(X_1+Y_1,R_1(X_0,Y_0))+R_2(X_0,Y_0)$S_2=R_0(X_2,Y_2)+R_1(X_1,Y_1)+R_1(R_0(X_1,Y_1),R_1(X_0,Y_0))+R_2(X_0,Y_0)$.

Can someone make the shape of $S_n$ more precise, e.g. in the form $S_n=P_0+\dots+P_n$ presumably with $P_0=R_0(X_n,Y_n)$, $P_n=R_n(X_0,Y_0)$? The intermediary $P_i$'s are more complicated but should be (uniquely) determined by a condition of the type
"$P_i$ is an iterated composition of $R_i$ involving only the variables $X_0,\dots,X_i$".
Maybe the polynomials $P_i$ should be homogeneous w.r.t. some graduation.

Any hint or relevant reference will be appreciated. Thanks!

1

# Polynomials for addition in the Witt vectors

The addition of $p$-typical Witt vectors ($p$ a prime number) is given by universal polynomials $S_n=S_n(X_0,\dots,X_n;Y_0,\dots,Y_n)\in\mathbb{Z}[X_0,X_1,\dots;Y_0,Y_1,\dots]$ determined by the equalities

$\Phi_n(S_0,\dots,S_n)=\Phi_n(X_0,\dots,X_n)+\Phi_n(Y_0,\dots,Y_n)$ for all $n\ge 0$,

where

$\Phi(T_0,\dots,T_n)=(T_0)^{p^n}+p(T_1)^{p^{n-1}}+\dots+p^nT_n$.

I guess that whoever sees the Witt vectors for the first time writes down explicitly $S_0=X_0+Y_0$, $S_1=X_1+Y_1+\frac{1}{p}((X_0)^p+(Y_0)^p-(X_0+Y_0)^p)$, maybe $S_2$ if she/he is courageous, and then stops since the computation becomes extremely messy. I think that there is no reasonable explicit expression in general, but patterns seem to exist and my question is about making these patterns more precise. Before I ask, let me illustrate with $S_2$. It is easy to see that there exists a unique sequence of polynomials $R_n\in\mathbb{Z}[X,Y]$, $n\ge 0$, such that

$X^{p^n}+Y^{p^n}=R_0(X,Y)^{p^n}+pR_1(X,Y)^{p^{n-1}}+\dots+p^nR_n(X,Y)$.

For example $R_0=X+Y$ and $R_1=\frac{1}{p}(X^p+Y^p-(X+Y)^p)$. Then:

$S_1=R_0(X_1,Y_1)+R_1(X_0,Y_0)$

$S_2=R_0(X_2,Y_2)+R_1(X_1,Y_1)+R_1(X_1+Y_1,R_1(X_0,Y_0))+R_2(X_0,Y_0)$.

Can someone make the shape of $S_n$ more precise, e.g. in the form $S_n=P_0+\dots+P_n$ presumably with $P_0=R_0(X_n,Y_n)$, $P_n=R_n(X_0,Y_0)$? The intermediary $P_i$'s are more complicated but should be (uniquely) determined by a condition of the type
"$P_i$ is an iterated composition of $R_i$ involving only the variables $X_0,\dots,X_i$".
Maybe the polynomials $P_i$ should be homogeneous w.r.t. some graduation.

Any hint or relevant reference will be appreciated. Thanks!