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Let $F(x,y)$ be a bivariate polynomial, of degree n. Hence:

$F(x,y) = \underset{i+j \leq n}{\sum_{i=0}\sum_{j=0}}a_{ij}x^{i}y^{j}$

Can there exist an upperbound for the number of isolated zeros for $F(x,y)$ ? I understand that if we ask for number of zeros in general, it can be infinite. But by isolated zeros I mean that zeros that are not connected by a curve (on the x-y plane).

I saw an answer to a similar question: Point 2 of this answer that one can bound the number of isolated zeros by $n^{2}$, or , it says some quadratic function of $n$. The answer was not elaborate to the point I could understand.

Can anyone kindly help me verify this ?

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    That answer also mentions "Harnack's Theorem." Did you try to find anything about it?2012-12-11
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    The paper at http://www.sciencedirect.com/science/article/pii/S0747717198902723 looks forbidding, but some of the references may give you what you need.2012-12-11
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    @GerryMyerson I looked at Harnack's Theorem for bivariate polynomials. It seems helpful, bounding the number of zero curves of the bivariate polynomial. Thanks for the reference, the paper seems heavy for me, but from what I understood, in it they are discussing solutions of a **polynomial system**, whereas I am looking for the **number of zeros of a single bivariate polynomial**. So, I was wondering it there are infact the same. Thanks again for the reference.2012-12-11
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    Well, a single polynomial is a system of $1$ polynomial; conversely, the system $f_1=f_2=\cdots=f_m=0$ has the same (real) zeros as the single equation $f_1^2+f_2^2+\cdots+f_m^2=0$. So it may be possible to use results about the one to get answers about the other.2012-12-11
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    Note also that if $f(x)$ has degree $n$ and $n$ distinct real zeros, then $(f(x))^2+(f(y))^2=0$ has degree $2n$ and $n^2$ isolated zeros.2012-12-11
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    Another thought: if $f(x,y)$ has an isolated zero at $p$, then it has a maximum or minimum there, so its partial derivatives are both zero. That's a system of two bivariate polynomials of degree at most $n-1$ each, so another link to polynomial systems.2012-12-12
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    @GerryMyerson Sorry, I didn't get why the function must have a minimum or a maximum at a zero ? I thought it could be increasing/decreasing or the zero can be an inflection point. But maybe I am missing the main point here.2012-12-12
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    However, this pdf http://arxiv.org/pdf/1102.5391.pdf seems quite helpful.2012-12-12
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    We are talking about a function of 2 variables with an isolated zero, right? Think of its graph: how can a zero be isolated, if there are both negative and positive values arbitrarily nearby? Anyway: if you find that you can answer your question after studying the link you give, you can write it up and post it as an answer here so we can all see it.2012-12-12

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