Schwartz-Zippel lemma and testing polynomial identities

Polynomial identity testing is the problem of determining whether a given multivariate polynomial is the0-polynomial or identically equal to 0. The input to the problem is an "n"-variable polynomial over a fieldF. It can occur in the following forms:

; Algebraic form : e.g. Is $(x\_1\; +\; 3x\_2\; -\; x\_3)(3x\_1\; +\; x\_4\; -\; 1)\; cdots\; (x\_7\; -\; x\_2)\; equiv\; 0$. To solve this, we can multiply it out and check that all the co-efficients are 0. However, this takes exponential time.; Determinant of a matrix with polynomial entries: Let $p(x\_1,x\_2,\; ldots,\; x\_n)$ be the determinant of the polynomial matrix.; Read-once branching program : also called binary decision diagrams.

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. The first of these algorithms was discovered independently by Schwartz and Zippel. [cite web
url=http://historical.ncstrl.org/tr/ps/cornellcs/TR89-965.ps
title= An Explicit Separation of Relativised Random Polynomial Time and Relativised Deterministic Polynomial Time
accessdate= 2008-06-15
author= Richard Zippel
year= 1989
month= February
format= ps]

Schwartz-Zippel theorem

The Schwartz-Zippel theorem is a tool commonly used in probabilistic polynomial identity testing. It bounds the probability that a non-zero polynomial will have roots at randomly selected test points. The formal statement is as follows:

Theorem 1 (Schwartz-Zippel). "Let $Pin\; F\; [x\_1,x\_2,ldots,x\_n]$ be a non-zero polynomial of degree $dgeq0$ over a field, $F$. Let $mathrm\{S\}$ be a finite subset of $F$ and let $r\_1,r\_2,ldots,r\_n$ be selected randomly from $S$. Then"

::: $Pr\; [P(r\_1,r\_2,ldots,r\_n)=0]\; leqfrac\{d\}$

If we denote the event $P(r\_1,r\_2,ldots,r\_n)=0$ by $A$ and the event $P\_i(r\_2,ldots,r\_n)=0$ by $B$, we have

=frac{d}

Applications

The importance of the Schwartz-Zippel Theorem and Testing Polynomial Identities followsfrom algorithms which are obtained to problems that can be reduced to the problemof polynomial identity testing.

Comparison of two polynomials

"Given a pair of polynomials $p\_1(x)$ and $p\_2(x)$, is" ::: $p\_1(x)\; equiv\; p\_2(x)$?

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

::: $[p\_1(x)\; -\; p\_2(x)]\; equiv\; 0$

Hence if we can determine that ::: $p(x)\; equiv\; 0$, where

::: $p(x)\; =\; p\_1(x);-;p\_2(x)$, then we can determine whether the two polynomials are equivalent.

Comparison of two polynomials (and therefore testing polynomial identities) also hasapplications in 2D-compression, where the problem of finding the equality of two2D-texts "A" and "B" is reduced to the problemof comparing equality of two polynomials $Poly\_A(x,y)$ and $Poly\_B(x,y)$

Primality testing

"Given $n\; in\; mathbb\{Z^+\}$, is $n$ a prime number?"

A simple randomized algorithm developed by Manindra Agrawal and Somenath Biswas can determine probabilisticallywhether $n$ is prime and uses polynomial identity testing to do so.

They propose that all prime numbers "n" (and only prime numbers) satisfy the followingpolynomial identity:

::: $(1+z)^n\; =\; 1+z^n\; (mbox\{mod\};n).$

This is a consequence of the Frobenius Homomorphism.

Let

::: $mathcal\{P\}\_n(z)\; =\; (1+z)^n\; -\; 1\; -z^n.,$

Then $mathcal\{P\}\_n(z)\; =\; 0;(mbox\{mod\};n)$ "iff n is prime". The proof can be found in [4] . However, since this polynomial has degree $n$, and since $n$ may or may not be a prime, the Schwartz-Zippel method would not work. Agrawal and Biswas use a more sophisticated technique, which divides $mathcal\{P\}\_n$ by a random monic polynomial of small degree.

Prime numbers are used in a number of applications such as hash table sizing, pseudorandom numbergenerators and in key generation for cryptography. Therefore finding very large prime numbers(on the order of $10^\{10,000\}$) becomes very important and efficient primality testing algorithmsare required.

Perfect matching

"Let $G\; =\; (V,\; E)$ be a graph of $mathrm\{n\}$ vertices where $mathrm\{n\}$ is even. Does $mathrm\{G\}$ contain a perfect matching?"

Theorem 2 (Tutte): "A Tutte matrix determinant is not a $mathrm\{0\}$-polynomial if and only if there exists a perfect matching."

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

:::

where

:::

The Tutte matrix determinant (in the variables "x_ij", "ideterminant of this skew-symmetric matrix which coincides with the square of the pfaffian of the matrix "A" and is non-zero (as polynomial) if and only if a perfect matching exists.One can then use polynomial identity testing to find whether $mathrm\{G\}$ contains a perfect matching.

In the special case of a balanced bipartite graph on $n\; =m\; +\; m$ 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.

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