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Two variables polynomial 
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#1
Dec1612, 11:46 AM

P: 206

Hi
Let p(x,y)≥0 be a polynomial of degree n such that p(x,y)=0 only for x=y=0.Does there exist a positive constant C such that the inequality p(x,y)≥C (IxI+IyI)^n (strong inequality!) holds for all 1≤x,y≤1? The simbol I I stands for absolute value. 


#2
Dec2212, 01:35 AM

P: 4,579

Hey hedipaldi.
What does (IxI + IyI)^n refer to? (what are the I's)? 


#3
Dec2212, 06:09 AM

P: 206

Thise are absolute values.It means [abs.val(x)+abs.val(y)]^n



#4
Dec2212, 05:58 PM

P: 4,579

Two variables polynomial
If (0,0) is the only root then it means that everything is greater than 0.
The only thing now is to consider the makeup of a 2D polynomial. If the double polynomial (or bivariate polynomial) has a structure p(x,y) = (a_n*x^n + a_(n1)*x^(n1) + ... + a0)*(b_n*y^n + b_(n1)*y^(n1) + ... + b0) and consider the behaviour in the region x, y <= 1. 


#5
Dec2212, 07:06 PM

P: 206

Does the limit p(x,y)/[absvalue(x)+absvalue(y)]^n nesecarily exist (finite or +infinite)?
This will solve my problem. 


#6
Dec2212, 07:24 PM

P: 4,579

What limit are you thinking of? (In other words what does x and y tend to)?



#7
Dec2212, 07:47 PM

P: 206

x and y tend to o. i.e (x,y) tends to (0,0).



#8
Dec2212, 07:55 PM

P: 4,579

It will tend to zero because all polynomials (including bivariate ones) are continuous.
Continuity implies that lim x>a, y>b f(x,y) = f(a,b) = 0 for (a=0,b=0). 


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