Register to reply

Two variables polynomial

by hedipaldi
Tags: polynomial, variables
Share this thread:
hedipaldi
#1
Dec16-12, 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.
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
chiro
#2
Dec22-12, 01:35 AM
P: 4,573
Hey hedipaldi.

What does (IxI + IyI)^n refer to? (what are the I's)?
hedipaldi
#3
Dec22-12, 06:09 AM
P: 206
Thise are absolute values.It means [abs.val(x)+abs.val(y)]^n

chiro
#4
Dec22-12, 05:58 PM
P: 4,573
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_(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
#5
Dec22-12, 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.
chiro
#6
Dec22-12, 07:24 PM
P: 4,573
What limit are you thinking of? (In other words what does x and y tend to)?
hedipaldi
#7
Dec22-12, 07:47 PM
P: 206
x and y tend to o. i.e (x,y) tends to (0,0).
chiro
#8
Dec22-12, 07:55 PM
P: 4,573
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).


Register to reply

Related Discussions
Complex Variables Polynomial Calculus & Beyond Homework 5
Polynomial equation in several variables Linear & Abstract Algebra 2
Polynomial with 3 unknown variables Precalculus Mathematics Homework 7
Complex conjugate variables as independent variables in polynomial equations Linear & Abstract Algebra 0