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.

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