Characterization of homogeneous polynomials over R

Question

I'm trying to prove the following:

Let $P(x_1, \dots, x_n) \in \mathbb{R}[x_1, \dots, x_n]$, and suppose that

$P(a_1, \dots, a_n) = 0 \implies P(ta_1, \dots, ta_n) = 0 \text{ } \text{ } \forall t \in \mathbb{R}$

Then $P(x_1, \dots, x_n)$ is homogeneous.

Thanks!

Answer 1

You can write $f$ as the sum of homogeneous polynomials, say $$f=f_d+f_{d-1}+\cdots+f_0$$ where $\deg f_m=m$, and $d=\deg f$. Now, denote $a=(a_1,\ldots,a_n)$, if $f(a)=0$, implies $f(ta)=0$, this is: $$f(ta)=t^df_d(a)+t^{d-1}f_{d-1}(a)+\cdots+tf_1(a)+f_0(a)=0$$ as this is zero for all $t$, then $f_n(a)=0$ for all $n$. This give us an idea for a counter example: For instance $g(x,y)=x^2+y^4$, if $g(a,b)=0$, then $a=b=0$ (we are on the reals) we have that $g(ta,tb)=t^2a^2+t^4b^4=0$ for all $t\in\mathbb{R}$.

So it is FALSE.