1. Statement of the lemma
     Algebraic form    Determinant of a matrix with polynomial entries  

2. Applications
     Comparison of two polynomials    Primality testing    Perfect matching  

3. Notes

4. References

5. External links

In mathematics, the Schwartz–Zippel lemma (also called the DeMillo-Lipton-Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomial is the

0-polynomial (or identically equal to 0). It was discovered independently by Jack Schwartz,^[1] Richard Zippel,^[2] and Richard DeMillo and Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result.^[3] The finite field version of this bound was proved by Øystein Ore in 1922.^[4]

Statement of the lemma

The input to the problem is an n-variable polynomial over a field F. It can occur in the following forms:

Algebraic form

For example, is

To solve this, we can multiply it out and check that all the coefficients are 0. However, this takes exponential time. In general, a polynomial can be algebraically represented by an arithmetic formula or circuit.

Determinant of a matrix with polynomial entries

Let

be the determinant of the polynomial matrix.

Currently, there is no known sub-exponential time algorithm that can solve this problem deterministically. However, there are randomized polynomial algorithms for testing polynomial identities. Their analysis usually requires a bound on the probability that a non-zero polynomial will have roots at randomly selected test points. The Schwartz–Zippel lemma provides this as follows:

In the single variable case, this follows directly from the fact that a polynomial of degree d can have no more than d roots. It seems logical, then, to think that a similar statement would hold for multivariable polynomials. This is, in fact, the case.

Proof. The proof is by mathematical induction on n. For {{math|1=n = 1}}, as was mentioned before, P can have at most d roots. This gives us the base case.

Now, assume that the theorem holds for all polynomials in {{math|n − 1}} variables. We can then consider P to be a polynomial in x₁ by writing it as

Since {{mvar|P}} is not identically 0, there is some {{mvar|i}} such that is not identically 0. Take the largest such {{mvar|i}}. Then , since the degree of is at most d.

Now we randomly pick from {{mvar|S}}. By the induction hypothesis,

If , then is of degree {{mvar|i}} (and thus not identically zero) so

If we denote the event by {{mvar|A}}, the event by {{mvar|B}}, and the complement of {{mvar|B}} by , we have

Applications

The importance of the Schwartz–Zippel Theorem and Testing Polynomial Identities follows

from algorithms which are obtained to problems that can be reduced to the problem

of polynomial identity testing.

Comparison of two polynomials

Given a pair of polynomials and , is

?

This problem can be solved by reducing it to the problem of polynomial identity testing. It is equivalent to checking if

Hence if we can determine that

where

then we can determine whether the two polynomials are equivalent.

Comparison of polynomials has applications for branching programs (also called binary decision diagrams). A read-once branching program can be represented by a multilinear polynomial which computes (over any field) on {0,1}-inputs the same Boolean function as the branching program, and two branching programs compute the same function if and only if the corresponding polynomials are equal. Thus, identity of Boolean functions computed by read-once branching programs can be reduced to polynomial identity testing.

Comparison of two polynomials (and therefore testing polynomial identities) also has

applications in 2D-compression, where the problem of finding the equality of two

2D-texts A and B is reduced to the problem

of comparing equality of two polynomials and .

Primality testing

Given , is a prime number?

A simple randomized algorithm developed by Manindra Agrawal and Somenath Biswas can determine probabilistically

whether is prime and uses polynomial identity testing to do so.

They propose that all prime numbers n (and only prime numbers) satisfy the following

polynomial identity:

This is a consequence of the Frobenius endomorphism.

Let

Then iff n is prime. The proof can be found in [4]. However,

since this polynomial has degree , and since may or may not be a prime,

the Schwartz–Zippel method would not work. Agrawal and Biswas use a more sophisticated technique, which divides

by a random monic polynomial of small degree.

Prime numbers are used in a number of applications such as hash table sizing, pseudorandom number

generators and in key generation for cryptography. Therefore, finding very large prime numbers

(on the order of (at least) ) becomes very important and efficient primality testing algorithms

are required.

Perfect matching

Let be a graph of {{math|n}} vertices where {{math|n}} is even. Does {{math|G}} contain a perfect matching?

A subset {{math|D}} of {{math|E}} is called a matching if each vertex in {{math|V}} is incident with at most one edge in {{math|D}}. A matching is perfect if each vertex in {{math|V}} has exactly one edge that is incident to it in {{math|D}}. Create a Tutte matrix {{math|A}} in the following way:

where

The Tutte matrix determinant (in the variables x_ij, {{tmath|iA and is non-zero (as polynomial) if and only if a perfect matching exists.

One can then use polynomial identity testing to find whether {{math|G}} contains a perfect matching. There exists a deterministic black-box algorithm for graphs with polynomially bounded permanents (Grigoriev & Karpinski 1987).^[5]

In the special case of a balanced bipartite graph on vertices this matrix takes the form of a block matrix

if the first m rows (resp. columns) are indexed with the first subset of the bipartition and the last m rows with the complementary subset. In this case the pfaffian coincides with the usual determinant of the m × m matrix X (up to sign). Here X is the Edmonds matrix.

Notes

References

• {{cite journal

• {{cite journal

• {{cite book|last2=Karpinski|first2=Marek|first1=Dima|last1=Grigoriev|title=The matching problem for bipartite graphs with polynomially bounded permanents is in NC|journal=Proceedings of the Annual Symposium on Foundations of Computer Science|year=1987|pages=166–172|doi=10.1109/SFCS.1987.56|ref=harv|isbn=978-0-8186-0807-0}}
• Moshkovitz, Dana (2010). An Alternative Proof of The Schwartz-Zippel Lemma. {{ECCC|2010|10|096}}
• {{cite journal

• {{cite book

• {{cite journal

• {{cite journal

• {{cite book

• {{cite web

• {{cite book

External links

• The Curious History of the Schwartz–Zippel Lemma, by Richard J. Lipton

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