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Repeat the random evaluation test for a polynomial identity with fresh independent random variables, keeping the same fixed polynomial. Reject as soon as any polynomial evaluation is nonzero; accept only if all $k$ values vanish. An identity is always accepted, preserving one-sided error. If a nonzero polynomial has single-trial false acceptance probability $p_1$, independence gives $p_k=p_1^k$. The Schwartz-Zippel lemma bounds this by $(d/|S|{)}^k$ when $d<|S|$.

For $G(x)=(x-1)(x-2)$ over the finite field ${\mathbb{F}}_5$, two uniform independent choices $(a,b)$ have $25$ equally likely ordered outcomes, each with probability $1/25$. Both polynomial evaluations are zero only at $(1,1),(1,2),(2,1),(2,2)$. Hence false acceptance falls from $2/5$ in one trial to $4/25$ in two trials, and correct rejection rises from $3/5$ to $21/25$.

If the second choice reuses the first, the pair is $(a,a)$ and only five diagonal outcomes are possible. Two still falsely accept, so the error remains $2/5$. Choosing fresh independent points is precisely what makes the full $25$-cell grid and the product of probabilities valid. For a target error $0<\epsilon <1$, this example needsThe ceiling function makes the trial count an integer. Four trials give $(2/5{)}^4=16/625>1/100$, while six give $64/15625<1/100$; five still give $32/3125>1/100$, so six is the first count below one percent. This is an exact error guarantee for each run on $G$, rather than an observed success rate.

A randomized decision algorithm has one-sided error when one answer is always correct and only the other answer can be mistaken.

The guarantee concerns each fixed input, with probability taken over the choices of the randomized algorithm. Always state which answer is protected. In polynomial identity testing, call an input positive when its polynomial is identically zero, and let acceptance mean claiming an identity. Then the possibilities are:

Thus a zero polynomial evaluation can leave uncertainty, but a nonzero polynomial evaluation settles the question. An error bound for a fixed nonzero polynomial is not the conditional probability that a particular accepted input is an identity; that latter question would need a distribution on the inputs as well.

In computational complexity theory, a polynomial-time algorithm with this orientation and false acceptance probability at most $1/2$ establishes membership in co-RP. Reversing the question to ask whether the polynomial is nonzero gives the RP orientation: negative instances are always rejected. The random evaluation test for a polynomial identity explains the required field, total degree, sampling and evaluation assumptions.

Polynomial identity testing asks whether a polynomial represented implicitly, such as by an arithmetic circuit or a remainder computation, is the zero polynomial.

Here zero means that every coefficient vanishes after collecting equal monomials in the polynomial ring. Testing equality of two polynomials $A,B$ amounts to testing $P=A-B$ for zero. An arithmetic circuit can permit cheap polynomial evaluation even when expanding all monomials is expensive. The random evaluation test for a polynomial identity exploits this: one nonzero value disproves the polynomial identity, while a known total degree limits how often a nonzero polynomial can evade detection.^[ref]

For example, for an integer $n\ge 1$, ${\prod }_{i=1}^n(1+x_i)$ has $2^n$ distinct monomials, one for each subset of the $n$ variables. Its factored expression needs only $n$ additions and $n-1$ multiplications for one polynomial evaluation. Evaluating the compact expression avoids creating its expanded list of coefficients. The saving is especially attractive in many variables; it does not make a zero result into a proof of identity.