Quadratic polynomial/R/2 variables/Pure form/List/Example

We make a list of real quadratic polynomials in the two variables ${}X$ and ${}Y$, together with their corresponding vanishing sets; we restrict to coefficients from ${}0,1,-1$. If only one variable ${}X$ occurs, then we have essentially the following three possibilities.

• ${}X^{2}\,$ ${}\,$ ${}\,$ the vanishing set is a "doubled line“.
• ${}X^{2}-1\,$ ${}\,$ ${}\,$ this means
${}X=\pm 1$, the vanishing set consists in two parallel lines.
• ${}X^{2}+1\,$ ${}\,$ ${}\,$ the vanishing set is leer.

In these cases (where the second variable ${}Y$ does not occur explicitly), the vanishing set is simply the product set of a zero-dimensional vanishing set (finitely many points) and of a line.

Now we consider polynomials where both variables occur.

• ${}Y^{2}-X\,$ ${}\,$ ${}\,$ the vanishing set is a parabola.
• ${}Y^{2}-X^{2}\,$ ${}\,$ ${}\,$ this means
${}(Y-X)(Y+X)=0$, the vanishing set consists in two lines crossing each other.
• ${}Y^{2}+X^{2}\,$ ${}\,$ ${}\,$ the only solution is the point ${}(0,0)$, the vanishing set is just a single point.
• ${}Y^{2}-X^{2}-1\,$ ${}\,$ ${}\,$ this means
${}(Y-X)(Y+X)=1$, the vanishing set is a hyperbola.
• ${}Y^{2}+X^{2}-1\,$ ${}\,$ ${}\,$ the vanishing sets is the unit circle.
• ${}Y^{2}+X^{2}+1\,$ ${}\,$ ${}\,$ again, this is empty.