Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
up vote 10 down vote accepted

No, consider $f(x,y)=1-xy^2$. $ $

share|cite|improve this answer
In that counterexample, the roots go to infinity when $x\to 0$. Which leads me to wonder what happens when the roots stay in a bounded interval, I updated my initial question – Ewan Delanoy Jul 27 '14 at 14:40
And it might have been preferable to ask another one. – Did Jul 27 '14 at 15:14
That’s a debatable issue, I myself have no opinion on the matter. I’ll put it into another question since you suggested it, but I predict that some other people will say : "it might have been preferable to stay in the same question". – Ewan Delanoy Jul 27 '14 at 15:40
I invented nothing about the matter, instructions on the site clearly suggest to do so (and common sense, I might add). Anyway, no big deal. – Did Jul 27 '14 at 17:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.