Symbol denoting "for at least one"

The existential quantifier "$\exists x.\phi(x)$" in formal logic denotes "there exists at least one $x$ that satisfies the property $\phi(x)$".

If you're not writing very formally symbolic logic, you should also consider sticking to English, but just writing "some" instead of "at least one", as in

Now, by the Fundamental Theorem of Algebra, $p(z)=0$ for some $z$. ...

since "at least one" is the conventional mathematical meaning of "some"+singular noun. With this wording you're even allowed to keep referring to the $z$ that makes $p(z)=0$ in the following sentences you write, whereas the $z$ in $\exists z.p(z)=0$ goes out of scope at the end of the formula.

Note that whereas one occasionally sees people abbreviate, for example, "$f(x)>0$ for all $x$" as "$f(x)>0 ~\forall x$", this heinous abuse of symbolism is rare for the existential quantifier. Properly used, quantifiers are always written before the formula they control: $$ \forall x.f(x)>0 $$ $$ \exists z.p(z)=0 $$ and so forth. Several minor variations in punctuation exist, such as $$ (\forall x)\, f(x)>0 $$ $$ \forall x(\,f(x)>0\,) $$