- #1
gvk
- 83
- 0
Does every positive polynomial in two real variables attain its lower bound in the plane? :yuck:
Do you mean the function which asymptoticaly aproaches the plane when x ->infinity?Hurkyl said:Let's start by investigating how it could fail.
Do you know of any way that a continuous function can fail to attain its lower bound?
A positive polynomial in two real variables is a polynomial expression with two variables (usually represented by x and y) that has only positive coefficients. This means that all the numbers in the polynomial are greater than zero.
The degree of a positive polynomial in two real variables is the highest power of the variables in the expression. For example, in the polynomial 3x^2 + 5xy + 2y^3, the degree is 3 because the highest power of any variable is 3.
A positive polynomial has only positive coefficients, while a non-positive polynomial can have both positive and negative coefficients. In other words, a positive polynomial will always result in a positive value when the variables are substituted, while a non-positive polynomial may result in a negative value.
No, a positive polynomial cannot have negative exponents. This is because a negative exponent would result in a fraction, and fractions cannot be considered positive. Therefore, all exponents in a positive polynomial must be zero or positive.
Positive polynomials are commonly used in optimization problems, where the goal is to find the maximum or minimum value of a function. They can also be used in economics, physics, and other fields to model relationships between variables and make predictions. Additionally, positive polynomials have applications in computer science, such as in graph theory and computer graphics.