2005/10/11 by Shujun Li, Li, Shujun
Computer Science · Mathematics · #11A07 #11C08 #11T06 #11Z05 #12E05 #13Fxx #Algebraic Geometry and Number Theory #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Rings and Algebras (math.RA) #math.NT #math.RA #msc:11A07 #msc:11C08 #msc:11T06 #msc:11Z05 #msc:12E05 #msc:13Fxx
paper · pdf · doi:10.48550/arxiv.math/0510217
26 pages
openalex publication_date 2005/10/11 · arxiv created 2005/11/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Zm=0, ..., m-1 generated by integer polynomials. The most well-known null polynomial is f(x)=xp-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 pd (d>=1). A typical monic null polynomial of the least degree modulo pd is given for any value of d>=1, from which one can further enumerate all null polynomials modulo pd. 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.