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Topic: Two-variable polynomials with a finite number of zeros
Replies: 9   Last Post: May 5, 2014 5:09 PM

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quasi

Posts: 10,232
Registered: 7/15/05
Re: Two-variable polynomials with a finite number of zeros
Posted: May 4, 2014 5:42 PM
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José Carlos Santos wrote:
>José Carlos Santos wrote:
>>
>>I suppose that if P(x,y) is a polynomial in two variables with
>>real coefficients which has only _n_ zeros, then the degree of
>>P(x,y) is greater than or equal to 2n. Am I right?

>
>No, I am wrong!
>
> (x^3 - x)^2 + (y^3 - y)^2
>
>has degree 6 and 9 zeros.


Which suggests ...

Conjecture:

If f in R[x,y] has exactly n zeros in R^2, then

(deg_x(f))*(deg_y(f)) >= 4n

where deg_x(f) denotes the degree of f in x, and deg_y(f)
denotes the degree of f in y.

quasi



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