How are polynomials called that are positive for all positive, real arguments, e.g., xy + z?


How can one determine if a polynomial has this property?



Positive polynomials
A matrix $A$ such that $x'Ax\geq 0$ for all $x$ is called positive semidefinite. A matrix $A$ such that $x'Ax\geq 0$ for all $x\geq 0$ (componentwise) is called copositive. While positive semidefiniteness is easy to test algorithmically, the weaker condition of copositivity is co-NP-complete to test.


Up to some details about strict vs. nonstrict inequalities which I do not think will be crucial, testing copositivity is an instance of the problem you propose, so we should expect this problem to be NP-hard. That is, we should not expect an efficient general procedure for solving it.


There are algorithmic techniques such as sums-of-squares and semidefinite programming which can in practice certify positivity of many polynomials (globally, or as you want here, on sets such as the positive orthant) quickly, but except in very special cases the efficiency of these methods is not guaranteed.


