$xy_1\in\langle x^2y_1,x^2y_2-xy_1,\dots,x^ny_n-xy_1\rangle$?

Question

Let $k$ be a field; let $n$ be an integer $\ge2$; let $x,y_1,\dots,y_n$ be indeterminates; and let $I$ be the ideal $$ \langle x^2y_1,x^2y_2-xy_1,\dots,x^ny_n-xy_1\rangle $$ in $k[x,y_1,\dots,y_n]$.

Do we have $xy_1\in I$?

Answer 1

Thank you very much to user26857 who explained in a comment that the answer is negative for the following reason:

Assume by contradiction $$ xy_1\in\langle x^2y_1,x^2y_2-xy_1,\dots,x^ny_n-xy_1\rangle. $$ Dividing by $x$ we get $$ y_1\in\langle xy_1,xy_2-y_1,x^2y_3-y_1,\dots,x^{n-2}y_{n-1}-y_1,x^{n-1}y_n-y_1\rangle. $$ Setting $y_i:=x^{n-i}y_n$ for $1\le i\le n-1$ we get $x^{n-1}y_n\in\langle x^ny_n\rangle$, which is false.