# Michael Artin’s Algebra Ch.3 M.3(c)

Problem Statement: prove that every pair $x(t), y(t)$ of real polynomials satisfies some real polynomial relation $f(x,y)=0$.

Solution sketch:

Let, without loss of generality, $x=a_0 + a_{1} t + a_{2} t^2+...+a_{n} t^n, a_{n} \neq 0,$ $y = b_0 + b_{1} t +...+ b_{m} t^m, b_{m} \neq 0, m \le n.$

Then $t^{m}$ can be written as a linear combination of  $y, t, t^{2},.., t^{m-1}$. So, in $x(t)$, we write $t^{k}$ as $t^{m}t^{k-m}$ when $m \le k$, and replace $t^{m}$ by $y - (b_0 + b_{1} t +...+ b_{m-1} t^{m-1}$. By repeating this process as long as there are powers of $t$ greater than or equal to $m$ in $x(t)$, we shall eventually find an equation where the highest power of $t$ is no greater than $m-1$. Now let the highest power of $t$ in this equation be $r$. Then in a similar fashion, we can plug in the value of $t^{r}$ in $y(t)$ to obtain an equation where the power of $t$ will not exceed $r - 1$. By repeating this process, we shall eventually obtain an equation where the power of $t$ is zero, and that is our desired $f(x,y)=0$.