Positivity of homogeneous form of the fourth degree

Question

I encountered an exercise that asked if the study of the positivity of

$Q(u,v) = a_0 u^4 + a_1 u^3 v + a_2 u^2 v^2 + a_3 u v^3 + a_4 v^4$

can be reduced to the study of the corresponding problem for quadratic forms. Can this be done?

Answer 1

I'm not sure i would put it that way; however, with real coefficients, your $Q$ factors as the product of two (homogeneous) quadratic forms with real coefficients. If the two factors are positive definite, you are finished. If both indefinite, you need to compare where they are negative; put another way, in this case $Q$ factors, again over the reals, as four linear terms $\beta u + \gamma v.$ If you have $$ \zeta \; \cdot (\beta u + \gamma v)^2 (\delta u + \epsilon v)^2$$ it is non-negative, else...

Well, final simple comment. If $v=0$ all that matters is $\alpha_0.$ For $v \neq 0,$ divide through by $v^4$ and call the result $f(r),$ where $r=u/v.$ So $$ f(r) = \alpha_0 r^4 + \alpha_1 r^3 + \alpha_2 r^2 + \alpha_3 r + \alpha_4. $$ Positivity of this is the same as positivity of the original. So, you can count the number of real roots, check for repeated roots with $\gcd(f(r),f'(r) )$ and so on.