What does it say about a multivariate polynomial to be zero on a linear subspace?

If I univariate polynomial $f(x)$ that vanishes at a point $x_0$, we conclude that $x - x_0$ divides $f(x)$, and in particular that $f$ is reducible if $\deg f > 1$. Can anything of significance be said if a multivariate polynomial $g : \mathbb{R}^d \to \mathbb{R}$ vanishes along an entire linear subspace $V < \mathbb{R}^d$?

The case $\dim V = 1$ can occur quite easily: if $g$ is homogenous, any zero of $g$ produces such a $V$. Thus, I am particularly interested in $\dim V = 2$ or higher, or if there is a taxonomy of different ways the $\dim V = 1$ case can occur.

This question is related to an earlier question on the Schwartz-Zippel lemma linked below. The motivation for both is that I have an algorithm that chooses a random linear subspace and succeeds whenever a particular multivariate polynomial doesn't vanish entirely on that subspace.

Analogue of the Schwartz–Zippel lemma for subspaces

If $V$ has codimension $1$ and is cut out by a single linear equation $\sum_{i=1}^d a_i x_i = 0$, then you can conclude that $\sum a_i x_i$ divides $f$. The proof is fairly short: by applying a suitable change of coordinates you can assume WLOG that the linear equation is $x_1 = 0$, and then it's clear.
In higher codimension you can't conclude much. For example, when $d = 2$ and $V$ has codimension $2$ then $V$ is a point, and vanishing on a point doesn't tell you too much in this case.
• Does anything interesting happen in the middle, say for $d = 4$ and $\dim V = 2$? May 28, 2013 at 4:28
• The general statement (which is not too hard to prove along the lines of the first statement) is that if $V$ is cut out by linear functionals $f_1, f_2, ... f_k$, then $f$ can be expressed as a sum $\sum f_i g_i$ where $g_i$ are some other polynomials. Is this interesting to you? I don't think it tells you much if $k > 1$. May 28, 2013 at 4:31
• That might be interesting, actually, if I prove some of my specific polynomials in question don't have that form. What do you specifically mean by "$V$ is cut out by linear functionals $f_1, \ldots, f_k$"? May 28, 2013 at 4:35
• To answer my own question, if $f$ is zero whenever $k$ linearly independent linear functionals $f_1, \ldots, f_k$ are zero, then after a coordinate change $f$ is zero whenever $x_1 = \cdots = x_k = 0$, and the desired result is immediate. May 28, 2013 at 4:56