Finding Singular points

Question

Let $f = 3x^3 + 3x^2 − y^2 + z^2$, $g = 3x^2 + 4^x + 3y^2 + z^2$ be polynomials in $\mathbb{C}[x, y, z]$ and let $W = V(\langle f, g\rangle) ⊂ \mathbb{A}^3(C)$. By using the Jacobian matrix, find the singular points of W or show that it has no singular points.(You may assume without proof that $\langle f, g\rangle$ is a prime ideal therefore $W$ is an irreducible variety and $I(W) = \langle f, g\rangle)$

I have $f_x=9x^2+6x$, $f_y=-2y$ and $f_z=2z$

and $g_x=6x+4$, $g_y=6y$, $g_z=2z$

so $J=\begin{bmatrix}9x^2+6x&-2y&2z\\6x+4&6y&2z\end{bmatrix}$

I don't know how to find the singular points (or show that it has none)

A quick search of stack exchange unearths several highly relevant posts which illustrate the methods you need (see here ). If you're looking for more specific guidance, please provide information about what you know and why you are confused (see "How to ask a good question" for general guidelines). — vociferous_rutabaga, Apr 20 '16 at 19:38

score 0 · Accepted Answer · answered Apr 21 '16 at 05:53

A point on $W$ is singular if and only if the rank of the Jacobian matrix at this point is $\leq 1$, which is (by basic linear algebra) equivalent to the fact, that all $2 \times 2$-minors vanish.

Hence you should compute the three $2 \times 2$-minors and see where they will vanish simultaneously.

Finding Singular points

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