How to know when a polynomial belongs to a certain ideal in $\mathbb{C}[x_1,x_2,x_3]$?

Question

I am trying to compute manually a Gröbner basis for $I=\langle f=x_3-x_1^5,g=x_2-x_1^3\rangle$ with the lexicographic order. After the third iteration I get, $$h_1=x_1^2x_2-x_3$$ $$h_2=x_1x_3-x_2^2$$ $$h_3=x_1x_2^3-x_3^2$$ $$h_4=x_3(x_1^4-x_2^3)$$ $$h_5=x_3^3-x_2^5$$ macaulay2 says that the basis should be $\{g,h_1,h_2,h_3,h_5\}$, so my question is, how can I prove that $h_4\in\langle g,h_1,h_2,h_3,h_5\rangle$ if the division algorithm doesn't work? This would be also usefull to prove that all the $S$-polynomials on the next iteration are 0.

Answer 1

I found my error computing $h_4$, $$h_4=S(f,h_3)=x_1^4x_3^2-x_2^3x_3$$ after that I used the division algorithm with $h_2$ and I get on the first iteration that, $$x_3(x_1^3x_3h_2-x_2^2g)=h_4$$ so it is, $$h_4\equiv 0\mod I$$