As a lazy mathematician, I'm tired of having to write "for at least one". Similar to the $\forall$ (for all) symbol, does a symbol exist to denote "for at least one"? I've been abbreviating to f.a.l.o., but I was hoping for some more elegant notation.
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 $$ (\forall x)\, f(x)>0 $$ $$ \forall x(\,f(x)>0\,) $$
Based on what I've read, it sounds like there is no standard abbreviation of "for at least one" or "for some", but what I typically use to denote "for some" in my own personal notes is the inverted F: $\Finv$ (LaTeX
It felt like a natural fit since it resembles the existential symbol $\exists$ (whose meaning it shares) just with the top line removed, and "F" is the first sound in the phrase "for some". I commonly use it when I wish to put an existential quantifier after what it quantifies over (improper as it may be), e.g. $$ x+1 = 2 \quad \Finv\ x $$ I considered using $\exists$ for this purpose, but that felt awkward since "for some" has no initial "E" sound, and it would be confusing to have to remember to read $\exists$ as "for some" only when it is placed after a formula.