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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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    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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