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 $z= 3x^2-5x$ .