Algebraic curves

To find the equations of an algebraic curve parametrized by $x=f(t),y=g(t),z=h(t)$, where $f,g,h$ are polynomials,

• Form the ideal $I\subset F[x,y,z,t]$ generated by $x-f(t)$, $y-g(t)$, $z-h(t)$.

• ``Project this ideal onto $F[x,y,z]$'': Compute the Gröbner basis of this ideal, using an ordering for which $x>y>z>t$. Choose only those terms in the basis whcih do not involve $t$.
• These terms will correspond to the equations for the curve in $s$ dimensions.

Example 2.8.13   The twisted cubic is parameterized by $x=t$, $y=t^2$, $z=t^3$. The object is to find a Gröbner basis for the ideal $I=(x-t,y-t^2,z-t^3)$ in $\mathbb{Q}[x,y,z,t]$. Let $<$ denote the lexicographic order defined by $t<z<y<x$. The Gröbner basis for $I$ is $G=\{y-x^2,z-x^3,-x+t\}$.

Therefore the algebraic equations for the curve are

$$y=x^2,\ z=x^3.$$

Example 2.8.14   Let $x=t^2+2t-5$, $y=t^4+4 t^3 +7 t^2 + 6 t-4$, $z=100 - 70 t - 23 t^2 + 12 t^3 + 3 t^4$ yields the ideal $I=( x - t^2-2t+5, y- t^4-4t^3-7t^2-6t+4, z - 100 + 70 t + 23 t^2 - 12 t^3 -3 t^4 )$ which has the Grobner basis $\{-32 - 13x - x^2 + y, 5 x - 3 x^2 + z, 5 - 2 t - t^2 + x\}$. So the equations for the curve are $y = x^2+13x+32$, and $z= 3x^2-5x$.