The limiting value, when the degree of the polynomial grows, is . In other words, for random polynomials of high degree and for large size fields, 36.79% of the time these polynomials will have a single root suitable for use as a signature.

If we have more than one linear factor for all the values
of *n* that we are prepared to try (notice that for
polynomials of degree 2 we either have two linear factors
or none), then we have multiple choices.
Whenever we have multiple choices we have to compute all
signatures arising from them.
So if we have a square root and an algebraic number which
factors in no less than 3 linear terms we will have to
compute
signatures.

If all the signatures are
the expression
is not zero.
If all *S*_{n}(*e*)=0 then the expression is identically
zero with some probability of error.
Else we cannot decide and the procedure fails.

Example.
Let and *p*(*x*) be defined as before.
The following is a standard normalization of
a rational expression containing algebraic numbers:

To compute the signature of these expressions, we need to find an

has a single linear factor and we decide to use it. So we set and

as expected.