Let $A$ be a $n \times n$ matrix over field $F$. Let $a\_1, \cdots, a\_n$ be the column vectors of $A$. For any subset $S \subseteq [n] = \{1, 2, \cdots, n\}$, let $a\_S = \sum\_{i \in S} a\_i$. Alon's celebrated [permanent lemma](http://lovelace.thi.informatik.uni-frankfurt.de/~jukna/EC_Book/exers_and_sols/node27.html) states that if the [permanent](http://en.wikipedia.org/wiki/Permanent) of $A$ is nonzero, then for any $b \in F^n$ there is some set $S$ such that $a\_S$ and $b$ differs in all coordinates.


Let $v\_S = \prod\_{i=1}^n (a\_{S,i} - b\_i)$. Then $v\_S \neq 0$ iff $a\_S$ and $b$ differ in all coordinates. By attempting to prove the lemma by my own, I discovered this identity.
$$\sum\_{S \subseteq [n]} (-1)^{n-|S|} v\_S = \operatorname{perm} A$$
So $\operatorname{perm} A \neq 0$ immediately implies the existence of some $v\_S \neq 0$. 
Note that the formula of left-hand side depends on $b$.


If we are working on a field with absolute value, then the absolute value of $v\_S$ measures the difference between $a\_S$ and $b$ in coordinates. For example, there is a set $S$ with $|v\_S| \geq (\operatorname{perm} A) / 2^n$. The bound is tight for $A = I\_n$ and $b = (1/2, \cdots, 1/2)$. This gives a quantitative version of permanent lemma.


Given the simplicity of the identity, I believe that it should had been noticed by other mathematicians. In what literature can I find observations about this identity? Are there any generalizations or applications around this idea?



About an identity which gives immediate proof of the permanent lemma
This identity is a particular case of theorem 3 in ["A generalization of Combinatorial Nullstellensatz"](http://arxiv.org/abs/1302.4647) by Michał Lasoń. Indeed your proof is essentially reinventing the combinatorial nullstellensatz. :)


The usual proof of Alon's lemma looks at the polynomial $\prod\_{i=1}^n (\sum\_j a\_{ij}x\_j - b\_i)$. And your identity follows by computing the coefficient of $x\_1x\_2\cdots x\_n$ in two different ways, like in the paper above.


