hwartz–zippel lemma

The Schwartz-Zippel lemma is a fundamental result in algebraic complexity theory that provides a way to bound the probability that a non-zero polynomial over a field evaluates to zero at a random point in that field. It states that if a polynomial P(x) of degree d in n variables over a field F is evaluated at a random point in F, then the probability that P(x) is zero is at most d/n. In other words, if the degree of the polynomial is small relative to the number of variables, then it is unlikely to evaluate to zero at a random point in the field. This result has many applications in algorithm design, including polynomial identity testing, randomized algorithms for solving systems of polynomial equations, and error-correcting codes.