2

We all know that, for polynomial functions of one real variable, say $x$, if zeros of polynomial $P$ are a subset of zeros of polynomial $Q$, then $P$ divides $Q$.

Assume that $P,Q$ are polynomials in several variables. For example, three: $P = P(x,y,z)$ and $Q = Q(x,y,z)$.

Does the property ($P(x,y,z)=0 \Rightarrow Q(x,y,z) =0$) imply $P$ divides $Q$?

PA6OTA
  • 1,972
  • 10
  • 12
  • For $P(x)=x^2$ and $Q(x)=x$, the zeros of $P$ are a subset of those of $Q$, yet it doesn't divide it. Did you mean something else? –  Nov 14 '16 at 22:23
  • @OpenBall It's possible that the OP meant counting multiplicities in the case of single variable polynomials (though indeed "subset of" is not the best way to describe that). – dxiv Nov 15 '16 at 07:47

1 Answers1

0

No, consider for example $P(x,y,z)=x^2+y^2+z^2$ and $Q(x,y,z)=x+y+z$.

dxiv
  • 76,497