Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$.
I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$.
In three variables, we have $6$ linearly independent monomials: $x^2,y^2,z^2,xy,xz,yz$ of degree $2$. Since $y\in I$, when we multiply them by $y$ we land in $I$, i.e. we find $$yx^2,y^3,yz^2,xy^2,xyz,y^2z\in I.$$ It seems to me that these stay linearly independent in the subspace $I\cap \mathbb C[x,y,z]_3$. Where am I wrong?