## Marie Litzinger

### A Basis for Residual Polynomials in n VariablesTransactions of the American Mathematical Society, Vol. 37, No. 2 (March 1935), 216-225.

Introduction

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 $$p_i(x)$$ which fulfills two requirements: (i) every residual polynomial modulo m is expressible as a sum of products of $$p_i(x)$$ polynomials in x with integral coefficients; (ii) no member of the set $$p_i(x)$$ 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 $$\mu(d)$$ denotes the least positive integer for which d divides $$\mu!$$. A special set of divisors of m is chosen: separate all divisors of m which exceed 1 into groups such that $$\mu(d)$$ 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, $$\Pi(\mu) = x(x-1)\dots(x-\mu+1);$$ when x is replaced by xj, the product will be denoted by $$\Pi_j(\mu)$$; $$\Pi(1)$$ 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 $$(m/d_i)\Pi(\mu(d_i))$$ for $$i=1,\dots,s$$ 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 $$p_i(x_1,\dots,x_n)$$ certain residual polynomials in several variables. This was brought to my attention by an example in two variables modulo 12. For this modulus, the $$d_1,\dots,d_s$$ are $$d_1=12$$, $$d_2=6$$, $$d_3=2$$; the corresponding $$\mu$$'s are $$\mu_1=4$$, $$\mu_2=3$$, $$\mu_3=2$$. Write $$q_i=m/d_i$$. The part of the basis containing one variable is composed of

(1)         12,  $$q_i\Pi_1(\mu_i)$$,   $$q_i\Pi_2(\mu_i)$$        (i = 1,2,3).

Kempner would include in the basis $$p_i(x_1,x_2)$$ modulo 12 only the seven terms (1). However, the residual polynomial,$P = (m/(d_3 \cdot d_3))\Pi_1(\mu_3)\Pi_2(\mu_3) = 3x_1(x_1-1)x_2(x_2-1)$must be added since, as is shown below, it is impossible to write the identity
(2)           $$\displaystyle P=12\cdot c + \sum_{i=1}^3q_i\Pi_1(\mu_i)f_i + \sum_{i=1}^3q_i\Pi_2(\mu_i)g_i$$
where $$c$$, $$f_i$$, $$g_i$$ are polynomials in $$x_1, x_2$$ with integral coefficients. By use of $$(x_1,x_2)$$ = (0,0), (2,0), (0,2) we prove the constant terms of $$c$$, $$f_3$$, $$g_3$$ even. The pair $$(x_1,x_2)$$ = (2,2) shows the right side of (2) divisible by 24 and the left side equal to 12.