I am looking for knowledge and intuition about the relation between shared roots and divisbility. More specifically, I am interested in statements in the following spirit:
Let $p$ and $q$ be two polynomial functions $\mathbb R^n\to\mathbb R$. If all roots of $p$ are also roots of $q$, then $p$ divides $q$ (there is a third polynomial $r$ so that $q=pr$).
This statement is not true as stated. Consider, for example $p(x)=1+|x|^2$ and $q(x)=1$ or these two multiplied by the same polynomial. My intuition seems to suggest that the only obstruction to this statement is bad behavior of $p$ (the only example I could think of is variants of having $1+|x|^2$ as a factor), whereas $q$ can be anything.
Is there a theorem or theory that explain to what extent the statement above is true? Does the statement hold for some "sufficiently nice" polynomials $p$? Does it hold in general modulo some kind of an error term in $q=pr$? I am specifically interested in it over the reals; the answer may well be different over an algebraically closed field. If there is no general theory or the general answer is a mess or even unknown, that would be useful to know as well.
In the case $n=1$ this is simpler. Assume that $p$ can be factored to first order terms. Then if a root of $p$ is a root of $q$ with at least equal multiplicity, then $p$ indeed divides $q$. In particular, the statement is true if $p$ has degree one. If $p$ has a second order irreducible factor (of which $1+x^2$ is a good example), then the statement fails. I am looking for this kind of understanding for $n\geq2$.
Think first about how this works with the factorisation of integers. $m=12$ and $n=18$ both have prime factors $2$ and $3$, but with different multiplicities. What we can say is that both are divisible by $2\times 3$ and that $12^2=18\times 8$ while $18^2=12\times 27$. We have that each divides a power of the other.
You can do the same with polynomials eg $p(x)=(x-2)^2(x-3)$ and $q(x)=(x-2)(x-3)^2$
So as well as taking care about what a root is, and working in an appropriate context to count all the roots, you also need to take care of multiplicities, even in very simple examples.