TITLE: Always a double root between "no roots" and "at least one root"?
QUESTION [5 upvotes]: Let $f(x,y)$ be a real bivariate polynomial. Suppose that $f(x,.)$
has no real roots when $x<0$, but has at least one real root when
$x>0$. Does it automatically follow that $f(0,.)$ has a double root ?

REPLY [10 votes]: No, consider $f(x,y)=1-xy^2$. $ $