What does the following interpretation of $k[x, y]/(y^2-x^3+x)$ mean? $k[x, y]/(y^2-x^3+x)$ is described as being "something like polynomial functions on the curve $y^2-x^3+x$."
I don't understand what this is supposed to mean. It is explained further that an important application of taking quotient rings is to find polynomials on some curve or other more complicated algebraic subset. Can you explain this some more also, please?
Reference: https://youtu.be/TBtG2IldX4M?t=1261
 A: You can view the polynomial ring $k[x,y]$ as the ring of polynomial functions on $k^2$ by sending a polynomial $f\in k[x,y]$ to the function
$$k^2 \rightarrow k, \, (a,b) \mapsto f(a,b).$$
Then two different polynomials also define two different polynomial functions, because you will always find points in $k^2$ where the two polynomials assign different values. But if you restrict to the curve $y^2 - x^3 +x$, i.e. to points $(a,b)$ in $k^2$ satisfying the equation $b^2 - a^3 + a = 0$, we have 'less points to check the value on a polynomial function on'. Then it can happen that two different polynomials in $k[x,y]$ define the same polynomial function on the curve.
For example, just take the polynomials $f = y^2 +x \in k[x,y]$ and $g = x^3\in k[x,y]$. These are clearly different polynomials, but for points $(a,b)$ on the curve we will always have
$$ f(a,b) = b^2 + a = a^3 = g(a,b),$$
so they give the same function!
The way to resolve this issue is to simply take the ring in which polynomials which will give the same function on the curve are the same. In other words, we mod out dependencies in the variables on curve. We then arrive at the ring $k[x,y]/(y^2 - x^3+x)$ that you described. This fix ensures that we once again have a one-to-one correspondence between elements of the ring and polynomial functions on our variety.
