Is there a standard notation for the set of roots of a polynomial

Question

For something so pervasive in mathematics, I'm surprised this doesn't have a simple and convenient notation. I suppose one could just always write $p^{-1}(0)$ but that somehow feels strange.

Answer 1

In algebraic geometry, a variety is defined to be the solution set of a system of polynomials over an algebraically closed field. More precisely, if $k$ is an algebraically closed field and $f_1,\dots,f_m$ are polynomials in $n$ variables with coefficients in $k$, then $$V(f_1,\dots,f_n) = \{(x_1,\dots,x_n) \in k^n \mid f_1(x_1,\dots x_n)=\cdots = f_m(x_1,\dots, x_n) = 0\}.$$ However, this does not necessarily coincide with what you are looking for, since I assume your $p$ is a map $\mathbb{R}\to \mathbb{R}$, and $\mathbb{R}$ is not algebraically closed. You could, however, write $V(p)$ to describe the set of solutions of $p$ in $\mathbb{C}$.