Let $A=\mathbb C[x_0,\dots,x_{m-1}]$ be the polynomial ring on $m$ variables.
Define $X(u)=\sum_{i=0}^{m-1} x_i u^{i+1}$ and denote by $(X(u)^r)_n$ the coefficient of $u^n$ in the expansion of the $r$-th power of $X(u)$, i.e $X(u)^r$.
Set $I=\langle (X(u)^{r})_s\mid 1\le r \le m+1, \ s\ge m+1 \rangle$.
I am trying to find a basis for $A/I$ and I am guided by some questions:
1) Is that possible to calculate the dimension of $A/I$? In fact, I am happy if I find a way to prove that it is at most $2^m$.
2) What is a Gröbner basis for $I$?
3) What is a linear basis for $A/I$?
Any help are welcome!
Added: If we let $m=2$, then $X(u) = x_0u +x_1u^2$ $X(u)^2 = x_0^2u^2 + 2x_0x_1u^3 + x_1^2 u^4$ $X(u)^3 = x_0^3 u^3+ 3x_0^2x_1u^4 + 3 x_0x_1^2u^5+ x_1^3u^6$ so then $I$ would be generated by $\{x_0x_1,\ x_1^2, \ x_0^3,\ x_0^2x_1, \ x_0x_1^2,\ x_1^3 \}.$ Therefore we conclude that $I$ is the ideal generated by $\{ x_0x_1,\ x_1^2, \ x_0^3 \}$, so that $A/I$ is a $\mathbb C$-space with basis given by the image (with respect to the natural projection) of $\{ 1, \ x_0, \ x_0^2, \ x_1 \}$.