Why is the zero polynomial the only one to have infinite roots?

Question

How can it be that the zero polynomial ($f(x)=0$) is the only polynomial which has an infinite number of roots? As stated on Wikipedia:

The polynomial $0$, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either $−1$ or $−∞$). These conventions are useful when defining Euclidean division of polynomials. The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots. The graph of the zero polynomial, $f(x) = 0$, is the $x$-axis.

We can have the polynomial $x-y$ to have infinitely many roots: $x=y=\text{all real numbers}$.

Where is my misunderstanding?

Answer 1

I would say that the statement you quote is ok, but it omits that it is only about polynomials of a single variable. For polynomials of several variables the statement is not true and you gave a good counterexample $P(x,y) = x-y$.

Actually, the study of the "roots" of polynomials of several variables is a huge field (the respective sets of roots are called "varieties" and they are studied in algebraic geometry).

Answer 2

The result in question is only about single-variable polynomials (hence the reference to "$f(x)$"). As $p(x,y)=x-y$ shows, a polynomial in more than one variable can indeed have infinitely many zeroes without being the zero polynomial.

Answer 3

Polynomial in a single variable can have only finite number of roots unless it is identically $0$. This does not extend to polynomials in several variables as your example shows.