Is There a Positive Constant for a Polynomial Inequality with Two Variables?

[hedipaldi]

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.

 

[chiro]

Hey hedipaldi.

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

 

[hedipaldi]

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

 

[chiro]

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_(n-1)*x^(n-1) + ... + a0)*(b_n*y^n + b_(n-1)*y^(n-1) + ... + b0) and consider the behaviour in the region |x|, |y| <= 1.

 

[hedipaldi]

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

 

[chiro]

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

 

[hedipaldi]

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

 

[chiro]

It will tend to zero because all polynomials (including bi-variate ones) are continuous.

Continuity implies that lim x->a, y->b f(x,y) = f(a,b) = 0 for (a=0,b=0).