# Every non-negative multivariate polynomial has even degree and the highest degree term has positive coefficient?

Part of my question has been asked before (Every non-negative multivariate polynomial has degree even?) but the proof there is not very satisfactory. The other part of my question involves proving (or disproving) that the highest-degree term, say $$c\prod_{i=1}^n x_i^{a_i}$$ (where we know $$\sum_i a_i$$ is even), must have its coefficient $$c > 0$$.

Are these two conditions (i.e., the polynomial having even degree and its highest degree term having positive coefficient) also sufficient for guaranteeing the polynomial is bounded from below?

Let $$R(x_1, \ldots, x_n)$$ be our polynomial, $$P(x_1, \ldots, x_n)$$ be a polynomial consisting of highest degree monomials from $$R$$ (it's homogeneous polynomial of odd degree) and $$Q = R - P$$.
We need that any polynomial with non-zero coefficients is non zero in at least one point. We can prove it by induction by number of variables: non-zero polynomials of $$1$$ variable have finite number of zeros; if polynomial of $$n + 1$$ variables is zero everywhere, then coefficients of any degree of the first variable are identically zero as polynomials of rest $$n$$ variables.
So for some $$a_i$$, $$P(a_1, a_2, \ldots, a_n) \neq 0$$. Thus $$P(a_1 t, a_2 t, \ldots, a_n t)$$ is non-zero homogeneous polynomial of one variable of odd degree. Then $$R(a_1 t, \ldots, a_n t)$$ is polynomial of odd degree - then it is negative at some point $$t_0$$, and so $$R$$ is negative at $$a_1 t_0, a_2 t_0, \ldots, a_n t_0$$.
For the second part - if highest degree term is unique and polynomial is positive - highest degree coefficient should be positive (otherwise polynomial will approach $$-\infty$$ as variables grow). It it's non unique - some coefficients can be negative. Take, for example, polynomial $$(x - y)^2 + 1$$.
Having even degree and all highest degree coefficients positive isn't enough for polynomial to be bounded: take, for example, $$x^2 - y$$.