In Wolfram (or maybe generally in many of textbooks of algebraic geometry), it's defined as;
Link: An algebraic curve over a field K is an equation f(X,Y)=0, where f(X,Y) is a polynomial in X and Y with coefficients in K.
I was wondering about the role of 0 in there. Let's say, we have f(x) = x. Then, to make it algebraic, we first introduce a variable y and move x on the same side as y, i.e., y - x = 0 and in the graph visualisation, this corresponds to converting (x, f(x)) w/h one variable into (x, y) with two variables.
So, can we say that 0 is there in the definition because of this operation? Any support is appreciated!
Yes, if you have an equation $g(x,y)=x$, you can write $f(x,y)=0$ where $f(x,y)=g(x,y)-x$. The definition of an algebraic curve is such that the points on the curve are the zeroes (often infinitely many) of $f(x,y)$. The other aspect of the definition is that an algebraic curve is a planar curve.