Computations of vanishing ideals

Question

I want to compute the following vanishing ideal:

For $V=V(x^2+y^2-1)\cap V(x-y)$ in the setting of $k[x,y]$ where $k$ is algebraically closed.

So we want to know $I(V(x^2+y^2-1)\cap V(x-y))$.

We can do this computation with basic properties and the Nullstellensatz.

$I(V(x^2+y^2-1)\cap V(x-y))=I(V(x^2+y^2-1, x-y))=\sqrt{(x^2+y^2-1, x-y)}$.

Doing some computations, we can find that $2y^2-1\in (x^2+y^2-1, x-y)$. I know looking up the theory of Gröbner Basis that $(2y^2-1, x-y)$ generates the ideal we take the radical of. It does not seem that we can simplify any further.

Is there an approach that does not use Gröbner Basis? Can we say more about this vanishing ideal?

Do you know an online tool that can compute radical ideals?

Thank you.

Answer 1

If the characteristic of $k$ is $2$, then $\newcommand{\V}{\mathbb{V}} x^2 + y^2 - 1 = (x+y+1)^2$, so \begin{align*} \V(x^2 + y^2 - 1) = \V((x+y+1)^2) = \V(x+y+1) \, . \end{align*} Then $$ V = \V(x+y+1) \cap \V(x+y) = \varnothing \, , $$ so $\newcommand{\I}{\mathbb{I}} \I(V) = k[x,y]$.

If the characteristic is not $2$, then \begin{align*} 2y^2-1 = x^2 + y^2 - 1 - (x+y)(x-y) \in (x^2+y^2-1, x-y) \end{align*} as you mentioned, and we have $(x^2+y^2-1, x-y) = (2y^2 - 1, x-y)$. Then \begin{align*} (x^2+y^2-1, x-y) &= (2y^2 - 1, x-y) = (y^2 - 1/2, x-y)\\ &= ((y-1/\sqrt{2})(y+1/\sqrt{2}), x-y)\\ &= (y-1/\sqrt{2}, x-y)(y+1/\sqrt{2}, x-y)\\ &= (x-1/\sqrt{2}, y-1/\sqrt{2})(x+1/\sqrt{2}, y+1/\sqrt{2}) \, . \end{align*} Since \begin{align*} 1 = -\frac{1}{\sqrt{2}}\left(\left(y-1/\sqrt{2}\right) - \left(y+1/\sqrt{2}\right) \right) \in (x-1/\sqrt{2}, y-1/\sqrt{2}) + (x+1/\sqrt{2}, y+1/\sqrt{2}) \, , \end{align*} then $(x-1/\sqrt{2}, y-1/\sqrt{2}) + (x+1/\sqrt{2}, y+1/\sqrt{2}) = k[x,y]$. The product of comaximal ideals is equal to their intersection (see here or here), so \begin{align*} (x^2+y^2-1, x-y) &= (x-1/\sqrt{2}, y-1/\sqrt{2})(x+1/\sqrt{2}, y+1/\sqrt{2})\\ &= (x-1/\sqrt{2}, y-1/\sqrt{2}) \cap (x+1/\sqrt{2}, y+1/\sqrt{2}) \, . \end{align*} Thus \begin{align*} \V(x^2+y^2-1, x-y) &= \V\left((x-1/\sqrt{2}, y-1/\sqrt{2}) \cap (x+1/\sqrt{2}, y+1/\sqrt{2})\right)\\ &= \V\left((x-1/\sqrt{2}, y-1/\sqrt{2})\right) \cup \V\left(x+1/\sqrt{2}, y+1/\sqrt{2})\right)\\ &=\{(1/\sqrt{2}, 1/\sqrt{2}), (-1/\sqrt{2}, -1/\sqrt{2})\} \, . \end{align*}

This agrees with the intuitive geometric picture: this set is the intersection of the unit circle with the line $y=x$.