Find $g\in I$ such that $LT(g)\notin \langle LT(g_1),LT(g_2),LT(g_3)\rangle$.

Question

Let $I=\langle g_1,g_2,g_3\rangle\subset \Bbb R[x,y,z]$ where $$g_1=xy^2-xy+y,\qquad g_2=xy-z^2, \text{ and } g_3=x-yz^4$$

Using lexicographic order find $g\in I$ such that $LT(g)\notin \langle LT(g_1),LT(g_2),LT(g_3)\rangle$

Here is my attempt

In lexicographic order we have $LT(g_1)=xy^2,\ LT(g_2)=xy,\ LT(g_3)=x$

Now the problem is finding a polynomial $g\in \Bbb R[x,y,z]$ such that $g$ can be expressed as a linear combination of $g_1,g_2,g_3$ but $LT(g)\notin\langle LT(g_1),LT(g_2),LT(g_3)\rangle$.

I guess $g=y^2$ which is not in $\langle LT(g_1),LT(g_2),LT(g_3)\rangle$. But I don't know how to express $y^2$ as a linear combination of $g_1,g_2,g_3$.

Answer 1

Set $g=g_1-yg_2+yg_3$. Then $g\in I$ and its leading term does not contain $x$.