Can a generating set for a polynomial ideal have less elements than a minimal Gröbner base?

Question

Let $\mathbb{K}$ be a field and $I\subset \mathbb{K}[x_1, \dots, x_n]$ an ideal. Then minimal Gröbner bases for $I$ (with respect to same monomial order) have the same number of elements, which is proven in e.g. https://www.kent.ac.uk/smsas/personal/gdb/MA574/week6.pdf. Let's call this number $k$.

Is it then possible that there exists a (non-Gröbner) generating set for $I$ with less than $k$ elements?

Answer 1

Let $I=\langle x^3-2xy, x^2y-2y^2+x \rangle$ be an ideal in $K[x,y]$ with grlex and $y\prec x$. Then a minimal Gröbner basis has $k=3$ elements, e.g., $$ G=\{ x^2,xy,y^2-\frac{1}{2}x \}. $$ But $I$ is generated by $2$ elements.