Kempner has established the existence of a basis for residual polynomials in one variable with respect to a composite modulus. A residual polynomial modulo m is by definition a polynomial f(x) with integer coefficients which is divisible by m for every integral value of x, and a residual congruence is written f(x) = 0 (mod m). By a basis for a given modulus is meant a finite set of residual polynomials pi(x) which fulfills two requirements: (i) every residual polynomial modulo m is expressible as a sum of products of pi(x) b polynomials in x with integral coefficients; (ii) no member of the set pi(x) can be written identically equal to a sum of products of the remaining members of the set by polynomials in x with integral coefficients.
For this work, the following notation is used. The symbol μ(d) denotes the least positive integer for which d divides μ!. A special set of divisors of m is chosen: separate all divisors of m which exceed 1 into groups such that μ(d) has the same value for all the d's of a group but different values for the d's of different groups; select the largest d of each group and denote this set by d1,..., ds. Finally, Π(μ) = x(x–1)...(x–μ+1); when x is replaced by xj, the product will be denoted by Πj(μ); Π(1) is interpreted as 1. Employing this notation, Dickson gave a brief proof of the theorem due to Kempner:
Every residual polynomial f(x) modulo m is a sum of products of m and (m/di)Π(μ(di)) for i = 1, ..., s by polynomials in x with integral coefficients.
In a later paper, Kempner considered the problem for n variables. In attempting to apply Dickson's method to the proof of the existence of a basis for residual polynomials in more than one variable, I found that Kempner had omitted from the set pi(x1,...,xn) certain residual polynomials in several variables. This was brought to my attention by an example in two variables modulo 12. For this modulus, the d1, ..., ds are d1=12, d2=6, d3=2; the corresponding μ's are μ1=4, μ2=3, μ3=2. Write qi = m/di. The part of the basis containing one variable is composed of
(1) 12, qiΠ1(μi), qiΠ2(μi) (i = 1,2,3).
Kempner would include in the basis pi(x1, x2) modulo 12 only the seven terms (1). However, the residual polynomial,
must be added since, as is shown below, it is impossible to write the identity
where c, fi, gi are polynomials in x1, x2 with integral coefficients. By use of (x1, x2) = (0,0), (2,0), (0,2) we prove the constant terms of c, f3, g3 even. The pair (x1, x2) = (2,2) shows the right side of (2) divisible by 24 and the left side equal to 12.