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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 ?

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up vote 10 down vote accepted

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

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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 at 14:40
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And it might have been preferable to ask another one. –  Did Jul 27 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 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 at 17:39

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