EUROCAL 1983: Computer Algebra pp 214-224

# Factorisation of sparse polynomials

• J. H. Davenport
Alsorithms 4 — Factorization
Part of the Lecture Notes in Computer Science book series (LNCS, volume 162)

## Abstract

Sparse polynomials xn±1 are often treated specially by the factorisation programs of computer algebra systems. We look at this, and ask how far this can be generalised. The answer is that more can be done for general binomials than is usually done, and recourse to a general purpose factoriser can be limited to "small" problems, but that general trinomials and denser polynomials seem to be a lost cause. We are concerned largely with the factorisation of univariate polynomials over the integers, being the simplest case.

## Keywords

Computer Algebra System Irreducible Polynomial Irreducible Factor Univariate Polynomial Factorise Algorithm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Aho, Hopcroft & Ullman,1974.
Aho, A.V., Hopcroft,J.E. & Ullman,J.D., The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.Google Scholar
2. Bogen et al.,1977
Bogen,R.A. et al., MACSYMA Reference Manual (version 9), M.I.T Laboratory for Computer Science, 1977.Google Scholar
3. Bremner,1981.
Bremner, A., On Reducibility of Trinomials. Glasgow Math. J., 22(1981), pp. 155–156. Zbl. 464.12001. MR 82i:10079.Google Scholar
4. Capelli,1898.
Capelli, A., Sulla Riduttibilità delle equazioni algebriche. Nota secunda. Rend. Accad. Sc. Fis. Mat. Soc. Napoli, Ser. 3, 4(1898), pp. 84–90.Google Scholar
5. Caviness,1975.
Caviness, B.F., More on Computing Roots of Integers. SIGSAM Bulletin 9(1975) 3, pp. 18–20, 29.Google Scholar
6. Davenport, Lewis & Schinzel,1961.
Davenport, H., Lewis, D.J. & Schinzel, A., Equations of the form f(x)=g(y). Quart. J. Math (Oxford) (2) 12(1961) pp. 102–104. H. Davenport, Collected Papers, Academic Press, 1977, Vol. IV, pp. 1711–1719.Google Scholar
7. Fried,1973.
Fried, M., The Field of Definition of Function Fields and a Problem in the Reducibility of Polynomials in Two Variables. Illinois J. Math. 17(1973) pp. 128–146.Google Scholar
8. Hearn,1973.
Hearn,A.C., REDUCE-2 User's Manual. Report UCP-19, University of Utah, 1973.Google Scholar
9. Knuth,1981.
Knuth,D.E., The Art of Computer Programming, vol. II, Seminumerical Algorithms (2nd. ed.). Addison-Wesley, 1981.Google Scholar
10. Lang,1970.
11. Lenstra et al., 1982
Lenstra,A.K., Lenstra,H.W.,Jr. & Lovász,L., Factoring Polynomials with Rational Coefficients. Preprint IW 195/82, Afdeling Informatica. Mathematisch Centrum, Amsterdam.Google Scholar
12. Ljunggren,1960.
Ljunggren, W., On the Irreducibility of Certain Trinomials and Quadrinomials. Math. Scand 8(1960) pp. 65–70.Google Scholar
13. Mikusiński & Schinzel,1964.
Mikusiński, J. & Schinzel, A., Sur la réductibilité de certains trinômes. Acta Arithmetica 9(1964) pp 91–95.Google Scholar
14. Moore & Norman,1981.
Moore, P.M.A. & Norman, A.C., Implementing a Polynomial Factorization and GCD Package. Proc. SYMSAC 81, ACM, New York, 1981, pp. 109–116.Google Scholar
15. Schinzel,1962.
Schinzel, A., Solution d'un problème de K. Zarankiewicz sur les suites de puissances consécutives de nombres irrationnels. Colloq. Math. 9(1962), pp. 291–296.Google Scholar
16. Schinzel,1963.
Schinzel, A., Some Unsolved Problems on Polynomials. Matematicka Biblioteka 25(1963) pp. 63–70.Google Scholar
17. Schinzel, 1973.
Schinzel, A., A General Irreducibility Criterion. J. Indian Math. Soc. (N.S.) 37(1973) pp. 1–8.Google Scholar
18. Schinzel, 1978.
Schinzel, A., Reducibility of lacunary polynomials III. Acta Arithmetica 34(1978) pp. 227–266.Google Scholar
19. Schinzel,1982.
Schinzel, A., Selected Topics on Polynomials. University of Michigan Press, Ann Arbor, Michigan, 1982.Google Scholar
20. Schinzel,1983.
21. Schnorr,1981.
Schnorr,C.P., Refined Analysis and Improvement On Some Factoring Algorithms. Proc. 8th. Colloquium on Automata, Languages and Programming (Springer Lecture Notes in Computer Science 115, 1981), pp. 1–15. Zbl. 469.68043.Google Scholar
22. Tverberg,1960.
Tverberg, H., On the Irreducibility of the Trinomials xn±xm±1. Math. Scand. 8(1960) pp. 121–126.Google Scholar
23. van der Waerden,1949.
van der Waerden, B.L., Modern Algebra, vol. I. Ungar, New York, 1949 (trans. from Moderne Algebra, 2nd. ed., Springer, 1937)Google Scholar
24. Vaughan,1974.
Vaughan, R.C., Bounds for the Coefficients of Cyclotomic Polynomials. Michigan Math. J. 21(1974), pp. 289–295.Google Scholar
25. Wang,1978.
Wang, P.S., An Improved Multivariable Polynomial Factorising Algorithm. Math. Comp. 32(1978) pp. 1215–1231. Zbl. 383.10035.Google Scholar
26. Zippel,1981.
Zippel, R.E., Newton's Iteration and the Sparse Hensel Algorithm. Proc. SYMSAC 81, ACM, New York, 1981, pp. 68–72.Google Scholar