In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic
polynomial identity testing In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and ...
. Identity testing is the problem of determining whether a given multivariate
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
is the
0-polynomial, the polynomial that ignores all its variables and always returns zero. The lemma states that evaluating a nonzero polynomial on inputs chosen randomly from a large-enough set is likely to find an input that produces a nonzero output.
it was discovered independently by
Jack Schwartz,
Richard Zippel, and
Richard DeMillo and
Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result. The finite field version of this bound was proved by
Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics.
Life
Ore graduated from the University of Oslo in 1922, with a ...
in 1922.
[Ö. Ore, Über höhere Kongruenzen. Norsk Mat. Forenings Skrifter Ser. I (1922), no. 7, 15 pages.]
Statement and proof of the lemma
Theorem 1 (Schwartz, Zippel). ''Let''
: