Exercise 5.7.3 (Curve passing through given points)

Proof. A general polynomial $f\in K[x,y]$ of degree $d$ in two variables is given by $f(x,y)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{(i,j)\in A}{a}_{i,j}{x}^i{y}^j$, where $A=\{(i,j){\text{∈}}^2\mid i+j\le d\}$. Here $K$ is any field, and $a_{i,j}\in K$.

Write $A_k=\{(i,j){\text{∈}}^2\mid i+j=k\}$. Then $A$ is the disjoint union of the $A_k$ for $0\le k\le d$, so that

Since $A_k=\{(0,k),(1,k-1),\dots ,(k,0)\}$, we obtain $\mathrm{Card}(A_k)=k+1$, thus

Consider $\left(\genfrac{}{}{0.0pt}{}{d+2}{2}\right)-1$ points $M_1,M_2,\dots ,M_{(d+1)(d+2)∕2-1}$ in the plane, where the coordinates of $M_k$ are $(x_k,y_k)$, without any condition concerning these points. Then the curve $f(x,y)=0$ passes through them if and only if

The unknowns $a_{\mathit{ij}}$ are solutions of this homogeneous linear system with $\left(\genfrac{}{}{0.0pt}{}{d+2}{2}\right)-1$ equations and $\left(\genfrac{}{}{0.0pt}{}{d+2}{2}\right)$ unknowns. Therefore there is a nontrivial solution of this system, so there exists a curve $f(x,y)=0$ of degree at most $d$ that passes through the points $M_k$.

If $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(f)<d$, we take $g=\mathit{fh}$, where $h$ is some polynomial with degree $d-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(f)$. Then the curve with equation $g(x,y)=0$ has degree $d$ and passes through the points $M_k$.

Example: Given nine points, there is some cubic which passes through these nine points. □