Null Polynomials modulo m

Abstract

This paper studies so-called "null polynomials modulo m", i.e., polynomials with integer coefficients that satisfy f(x)=0 (mod m) for any integer x. The study on null polynomials is helpful to reduce congruences of higher degrees modulo m and to enumerate equivalent polynomial functions modulo m, i.e., functions over Z_m={0, ..., m-1} generated by integer polynomials. The most well-known null polynomial is f(x)=x^p-x modulo a prime p. After pointing out that null polynomials modulo a composite can be studied by handling null polynomials modulo each prime power, this paper mainly focuses on null polynomials modulo p^d (d>=1). A typical monic null polynomial of the least degree modulo p^d is given for any value of d>=1, from which one can further enumerate all null polynomials modulo p^d. The most useful result obtained in this paper are Theorem 32 in Sec. 4.4 and its derivative -- Theorem 34 in Sec. 4.5. The results given in Sec. 4.3 form a basis of the induction proofs given in Sec. 4.4. However, if you do not care how the proofs in Sec. 4.4 were established, you can simply skip Sec. 4.3. Theorems 28 and 31 are very important for the proof of Theorem 32 and should be paid more attention. Note: After finishing this draft, we noticed that some results given in this paper have been covered in Kempner's papers [3,4]. Since we use a different way to obtain the results, this work can be considered as an independent and different proof. For a brief introduction to Kempner's proof, see the Appendix of this paper.