Proposition

Consider the equations

\[\begin{align*} x^2 + y^2 - 1 &= 0, \\ xy - 1 &= 0 \end{align*}\]

which describe the intersection of a circle and a hyperbola.

  1. Use algebra to eliminate $y$ from the above equations.
  2. Show how the polynomial found in part (1) lies in $\ev{ x^2 + y^2 - 1, xy - 1 }$.

Solution

1

Since $y = 1/x$, we have $x^4 - x^2 + 1 = 0$.

2

$x^4 - x^2 + 1 = x^2(x^2 + y^2 - 1) - (xy + 1)(xy - 1) \in \ev{x^2 + y^2 - 1, xy - 1}$.