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As I see it, the polynomial method is not limited to applications of the Combinatorial Nullstellensatz or any other specific result (as the Schwartz-Zippel lemma). Known for several decades at least, this method involves encoding combinatorial problems in fields (more generally rings, or even generic abelian groups) in terms of (non)vanishing of some polynomials, and then studying the resulting polynomial problem using various tools -- such as CN, SZ, and so on. One common theme (but certainly not exhausting the whole subject) is showing that a set with some particular combinatorial property is large: if it were small, we could construct a low-degree polynomial vanishing on this set (or its cartesian power), while such polynomial cannot exist by virtue of the combinatorial property under consideration.

As Fedor mentioned, this method usually works when we have a sharp result to prove, although there are some exceptions: say, the best up-to-date results on the finite fields Kakeya problem, obtained using the polynomial method, are unlikely to be sharp.

Anyway, absolutely vital is seems to be our ability to express the problem in terms of (non)vanishing of some polynomial.

Two more comments. First, it has been observed very recently that in the Combinatorial Nullstellensatz, one does not need $x_1^{t_1}\dots x_n^{t_n}$ to be a monomial of the largest possible degree; it suffices that it is not majorated by any other monomial.

Second, one does not have to confine to just vanishing: a very fruitful approach, to my knowledge first suggested by Saraf and Sudan and then further developed in their joint paper with Dvir and Kopparty, is to take into account the multiplicity with which a polynomial vanishes.

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As I see it, the polynomial method is not limited to applications of the Combinatorial Nullstellensatz or any other specific result (as the Schwartz-Zippel lemma). This method, known Known for several decades at least, consists in this method involves encoding combinatorial problems in fields (more generally rings, or even generic abelian groups) in terms of (non)vanishing of some polynomials, and then studying the resulting polynomial problem using various tools -- such as CN, SZ, and so on. One common theme (but certainly not exhausting the whole subject) is to show showing that a set with some particular combinatorial property is large: if it were small, we could construct a small-degree low-degree polynomial vanishing on this set (or its cartesian power), while such polynomial cannot exist by virtue of the combinatorial property under consideration.

As Fedor mentioned, this method usually works when we have a sharp result to prove, although there are some exceptions: say, the best up-to-date results on the finite fields Kakeya problem, obtained using the polynomial method, are unlikely to be sharp.

Anyway, absolutely vital seems to be the possibility is our ability to encode express the problem in terms of (non)vanishing of some polynomial.

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As I see it, the polynomial method is not limited to applications of the Combinatorial Nullstellensatz or any other specific result (as the Schwartz-Zippel lemma). This method, known for several decades at least, consists in encoding combinatorial problems in fields (more generally rings, or even generic abelian groups) in terms of (non)vanishing of some polynomials, and then studying the resulting polynomial problem using various tools -- such as CN, SZ, and so on. One common theme (but certainly not exhausting the whole subject) is to show that a set with some particular combinatorial property is large: if it were small, we could construct a small-degree polynomial vanishing on this set (or its cartesian power), while such polynomial cannot exist by virtue of the combinatorial property under consideration.

As Fedor mentioned, this method usually works when we have a sharp result to prove, although there are some exceptions: say, the best up-to-date results on the finite fields Kakeya problem, obtained using the polynomial method, are unlikely to be sharp.

Anyway, absolutely vital seems to be the possibility to encode the problem in terms of (non)vanishing of some polynomial.