Gröbner Basis for Ideal $J$

Question

I have the following question...

Consider the ideal $J:= (x^2y-x^2y^2,\ x^2z-z^2yx,\ x^2+xz) \subset \mathbb{Q}[x,y,z]$

Is $x \in J?$ Is $x \in \sqrt{J} $?

I know finding if $x$ is in the radical of $J$ I need to compute the Gröbner basis and work it out from there... But I don't know the method to use to show $x \in J $

Answer 1

Groebner basis for $J$ in lexicographic ordering $x \prec y \prec z$ is $$ J := \left(x z^2, x y z(y-1), x(x+z) \right) $$ By applying the multivariate division algorithm relative to the same ordering you find $$ x = 0 \cdot p_1 + 0 \cdot p_2 + 0 \cdot p_3 + x $$ therefore $x\not\in J$, but $$ x^3 = 1 \cdot p_1 + 0 \cdot p_2 + (x-z) \cdot p_3 + 0 $$ hence $x \in \operatorname{Rad}(J)$.

Here are relevant Mathematica commands:

In[9]:= gb = 
 GroebnerBasis[{x^2 y - x^2 y^2, x^2 z - z^2 x y, x^2 + x z}, {x, y, 
   z}, MonomialOrder -> Lexicographic]

Out[9]= {x z^2, -x y z + x y^2 z, x^2 + x z}

In[10]:= Table[PolynomialReduce[x^n, gb, {x, y, z}], {n, 1, 3}]

Out[10]= {{{0, 0, 0}, x}, {{0, 0, 1}, -x z}, {{1, 0, x - z}, 0}}