TITLE: Smallest $S\subset \mathbb C$ on which no degree $k$ polynomial always vanishes?
QUESTION [21 upvotes]: Say $p$ is a polynomial of degree $k$ in $\mathbb C[x]$.  Then $p$ can have at most $k$ distinct roots.  A somewhat obtuse way to state that is to say that among any set of $k+1$ distinct complex numbers, there must exist a value $a$ for which $p(a)\neq 0$.
The question here has to do with generalizing the above fact to multivariate polynomials.
Given some $p \in \mathbb C[x_1,\ldots,x_n]$, it turns out that  whenever we have enough different values to choose from, we can always pick from these some $a_1,\ldots,a_n$ so as to give $p(a_1,\ldots,a_n)\neq 0$.  Moreover, how many is "enough" is a function just of $n$ and the degree of $p$.
I want to emphasize that, in the above and in what follows, the $a_1,\ldots,a_n$ so chosen are distinct.
To make things more precise and specific, we have the following.

Theorem.  Let $p \in \mathbb C[x_1,\ldots,x_n]$ be a polynomial of total degree $k$.  Then any set of $n+k$ distinct complex numbers contains a subset of $n$ that can be assigned to the variables $x_1,\ldots,x_n$ so as to give a nonzero value for $p$.  (Again, we are talking about choosing from the set $n$ distinct values that we assign to $x_1,\ldots,x_n$.)

The proof of the above is a not-too-difficult exercise.
But the question is:

Question.  In the above theorem, can the $n+k$ be replaced with some other function of $n$ and $k$ so as to give a better upper bound?

In fact, let's define $M(n,k)$ to be the smallest positive integer such that for every $p\in \mathbb C[x_1,\ldots,x_n]$ of total degree $k$, every set $S$ of at least $M(n,k)$ distinct complex numbers contains some subset $\{a_1,\ldots,a_n\}$ (of cardinality $n$, so that these are distinct values) such that $p(a_1,\ldots,a_n)\neq 0$.
The theorem above states that $M(n,k) \le n+k$, but I don't even see how to meet that bound for $n=k=2$.

REPLY [11 votes]: This is probably just another way to present Fedor Petrov's solution: Expand
$$\frac{(1-t \alpha_1) (1-t \alpha_2) \cdots (1-t \alpha_{n+k-1})}{(1-t \beta_1)(1 - t \beta_2) \cdots (1-t \beta_n)}$$
as a formal power series in $t$. The coefficient of $t^k$ is a degree $k$ polynomial in the $\beta$'s, which vanishes whenever $\{ \beta_1, \beta_2, \ldots, \beta_n \}$ is a $n$-element subset of the $\alpha$'s.