This is a (self-contained) followup question to <https://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces>.


Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero multivariate polynomial of total degree $d$ over a field $\mathbb{R}$, and $S \subset \mathbb{R}$ be finite. Pick a positive integer $k$, choose $y\_1, \ldots, y\_k$ randomly and uniformly from $S^n$, and consider the $k$-variable polynomial


$$g(t\_1,\ldots,t\_k) = f(t\_1 y\_1 + \cdots + t\_k y\_k)$$


We seek an upper bound on the probability that $g(t)$ is the zero polynomial. Setting $t\_i = \delta\_{ij}$ for each $j$ and applying the Schwartz-Zippel lemma, we have


$$\Pr(g(t)=0) \le \frac{d^k}{|S|^k}$$


However, this bound does not use the fact that we have an entire subspace on which to be nonzero.


**Question 1**: Is there a slightly stronger unconditional bound when $k > 1$?


My reason for hope here is that the Schwartz-Zippel bound is achieved by a univariate polynomial with $d$ distinct roots, but in the subspace case $g(t) = 0$ only if we pick the same root for all $y\_i$, giving
$$\Pr(g(t)=0) \le \frac{d}{|S|^k}$$
if $f(x)$ depends on only one variable. In this case, we might save up to a factor of $d^{k-1}$.


**Question 2**: Say $f(x)$ is irreducible over $\mathbb{R}$, and we condition on $y\_i$ being full rank. Is there an even stronger bound when $d > 1$, ideally growing in strength with $n$?


**Edit**: Question 2 is false as stated, because $y\_i$ might be full rank but lower rank when projected onto the variables on which $f(x)$ depends. I am still hoping it holds if a suitable "depends on all variables" condition is added, but am not sure how to phrase this.



Stronger versions of Schwartz-Zippel for random linear subspaces
For question 1, I think it's enough to take a fixed $z$ that is transcendental over $\mathbb{Q}(S)$, and consider $$g(z,z^2,\dots,z^k)=f(zy\_1+z^2y\_2+\dots+z^k y\_k).$$


Each coordinate of $zy\_1+\dots+z^ky\_k$ is independently uniform on
$$S'=\{zs\_1+\dots+z^k s\_k \, | \, s\_1, \dots, s\_k \in S\},$$
so by Schwarz-Zippel we have 
$$P(g(z,z^2,\dots,z^k)=0) \leq \frac{d}{|S'|}=\frac{d}{|S|^k}$$


