In the post Balls in Bins I wrote about a combinatorial function which denotes the minimum value of the product among all distributions of balls (so ) in bins with the constraints . It turns out that this combinatorial function is linked to zeros of a multivariable polynomial in a “finite grid”. Here, by a finite grid I mean a subset of the form where is a field and ‘s are finite subsets of . The Alon-Furedi theorem, which provides this link, is as follows:
Theorem 1 (Alon-Furedi). Let be a finite grid in where . Let be an -variable polynomial of degree such that the set is nonempty (in other words, does not vanish on the entire grid). Then . Moreover, the bound is sharp.
For the sharpness of this bound let be any feasible distribution of balls in bins. Pick subsets with and define which has degree and doesn’t vanish on points of the grid .
This theorem appears in the last pages of  as Theorem 5, and until the appearance of  where Clark, Forrow and Schmitt used it to obtain a generalization of Warning’s second theorem and several interesting results in combinatorics and additive number theory, it seems like this result was largely ignored. In  Pete further generalized the results of  and gave some nice applications to graph theory, additive group theory and polynomial interpolation. In fact, he also generalized the Alon-Füredi theorem by replacing the field by a commutative ring and imposing some extra conditions on the grid which are vacuously true for the field case. While discussing some results of  with Pete and John I realised that we can do much more with this Alon-Furedi theorem, and this resulted in my joint paper with Pete, Aditya and John , which I have also discussed here. A particular contribution of  is the following generalization of Alon-Furedi (Theorem 3 below) for which we first need some motivation.
The famous Schwartz-Zippel lemma says that a polynomial in of degree has at most zeros in . In fact it works over more general grids as well where it says that if are finite subsets of with and a polynomial of degree , then has at most zeros in . Since Theorem 1 gives a lower bound on the number of non-zeros of a polynomial in a finite grid, it automatically gives an upper bound on the numbers of zeros of that polynomial. This observation leads to a natural question.
When the degree of the polynomial is at most , does Theorem 1 imply the Schwartz-Zippel lemma?
If you do the math, then you would see that indeed, it does. So, the Alon-Füredi theorem is in fact a generalization of the so-called Schwartz-Zippel lemma (see the blog post by RJ Lipton and some comments on it for why I have used “so-called” in this sentence). But, there are other results in the spirit of Schwartz-Zippel lemma, which do not follow from Theorem 1! The following result is due to DeMillo-Lipton and Zippel:
Theorem 2. Let be a non-zero polynomial such that for all , we have for some constant . Then for a subset of cardinality at least , the number of zeros of in is at most .
To see that this result doesn’t follow from Theorem 1, take and . Then Theorem 1 tells us that has at most zeros in , while Theorem 2 tells us that has at most zeros (which is stronger and the sharp bound in this case). This doesn’t mean that DeMillo-Lipton/Zippel’s result is always better. In fact, if you take , then Theorem 1 gives the better bound. To “rectify” this situation was one of our main motivation to give the following generalization of Theorem 1, which appears as Theorem 1.2 in .
Theorem 3 (Generalized Alon-Furedi). Let be a finite grid in where . For let be an integer such that . Let be an -variable polynomial of degree such that the set is nonempty and for all . Then we have . Moreover, the bound is sharp.
Here is defined to be the minimum value of the product among all distributions of balls in bins with the constraints . If you “do the math” (see Section 4 of ), then Theorem 3 is a common generalization of Theorem 1, Schwartz-Zippel lemma and Theorem 2.
 N. Alon and Z. Füredi, Covering the cube by affine hyperplanes. Eur. J. Comb. 14 (1993), 79–83.
 A. Bishnoi, P. L. Clark, A. Potukuchi, J. R. Schmitt, On Zeros of a Polynomial in a Finite Grid (2015). arXiv preprint arXiv:1508.06020.
 P. L. Clark, Fattening up Warning’s Second Theorem (2015). arXiv preprint arXiv:1506.06743.
 P. L. Clark, A. Forrow and J. R. Schmitt, Warning’s Second Theorem With Restricted Variables. To appear in Combinatorica, (2016).