Absolute irreducibility of polynomials via Newton polytopes

Journal of Algebra

Volumn 237, Issue 2, 2001, Pages 501-520

  Gao, Shuhong ^a  

^a CLEMSON UNIVERSITY   (United States)

Author keywords

Absolute irreducibility; Eisenstein criterion; Fields; Indecomposable polytopes; Minkowski sums; Multivariable polynomials; Newton polygons; Newton polytopes

Indexed keywords

EID: 0035869860     PISSN: 00218693     EISSN: None     Source Type: Journal    
DOI: 10.1006/jabr.2000.8586     Document Type: Article

References (52)

1. Adleman L. M. The function field sieve. Algorithmic Number Theory (Ithaca, NY, 1994). Lecture Notes in Comput. Sci. 877:1994;Springer-Verlag, Berlin. p. 108-121.
2. Balinski M. On the graph structure of convex polyhedra in n-space. Pacific J. Math. 11:1961;431-434.
3. Cassels J. W. S. Local Fields. London Mathematical Society Student Texts. 3:1986;Cambridge Univ. Press, London.
4. Dumas G. Sur queques cas d'irreductibilite des polynomes à coefficients rationnnels. J. Math. Pures Appl. 2:1906;191-258.
5. Eisenstein G. Über die Irreduzibilität und einige andere Eigenschaften der Gleichungen. J. Reine Angew. Math. 39:1850;160-179.
6. Ewald G. Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics. 168:1996;Springer-Verlag, Berlin/New York.
7. Filaseta M. The irreducibility of all but finitely many Bessel polynomials. Acta Math. 174:1995;383-397.
8. D. Gale, Irreducible convex sets, in "Proc. Internat. Congr. Math., Amsterdam, 1954," Vol. 2, pp. 217-218.
9. S. Gao, Factoring Multivariable Polynomials Via Differential Equations, preprint 2000.
10. Gao S., Shokrollahi M. A. Computing roots of polynomials over function fields of curves. Joyner D. Coding Theory and Cryptography: From Enigma and Geheimschreiber to Quantum Theory. 2000;214-228 Springer-Verlag, Berlin.
11. von zur Gathen J. Irreducibility of multivariate polynomials. J. Comput. System Sci. 31:1985;225-264.
12. Gel'fand I., Kapranov M., Zelevinsky A. Discriminants, Resultants and Multi-Dimensional Determinants. 1994;Birkhäuser, Boston.
13. Grünbaum B. Convex Polytopes. 1967;Interscience, London/New York/Sydney.
14. Hirschfeld J. W. P. Projective Geometries over Finite Fields. 1979;Clarendon, Oxford.
15. Janwa H., McGuire G., Wilson R. M. Double-error-correcting cyclic codes and absolutely irreducible polynomials over GF(2). J. Algebra. 178:1995;665-676.
16. Kallay M. Indecomposable polytopes. Israel J. Math. 41:1982;235-243.
17. Kaltofen E. Fast parallel absolute irreducibility testing. J. Symbolic Comput. 1:1985;57-67.
18. Kaltofen E. Deterministic irreducibility testing of polynomials over large finite fields. J. Symbolic Comput. 4:1987;77-82.
19. Kaltofen E. Effective Noether irreducibility forms and applications. Symposium on the Theory of Computing (New Orleans, LA, 1991). J. Comput. System Sci. 50:1995;274-295.
20. Khanduja S. K., Saha J. On a generalization of Eisenstein's irreducibility criterion. Mathematika. 44:1997;37-41.
21. Khovanskiǐ A. G. Fewnomials. Translations of Mathematical Monographs. 88:1991;Am. Math. Soc. Providence.
22. Kurschak J. Irreduzible formen. J. Reine Angew. Math. 152:1923;180-191.
23. Lidl R., Niederreiter H. Finite Fields. Encyclopedia of Math. and Its Appl. 20:1983;Addison-Wesley, Reading.
24. Lipkovski A. Newton polyhedra and irreducibility. Math. Z. 199:1988;119-127.
25. MacLane S. The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals. Trans. Amer. Math. Soc. 43:1938;226-239.
26. McMullen P. Indecomposable convex polytopes. Israel J. Math. 58:1987;321-323.
27. Meyer W. Indecomposable polytopes. Trans. Amer. Math. Soc. 190:1974;77-86.
28. Mott J. Eisenstein-type irreducibility criteria. Zero-Dimensional Commutative Rings (Knoxville, TN, 1994). Lecture Notes in Pure and Appl. Math. 171:1995;Dekker, New York. p. 307-329.
29. Noether E. Ein algebraisches Kriterium für absolute Irreduzibilität. Math. Ann. 85:1922;26-33.
30. Ore O. Zur Theorie der Irreduzibilitätskriterien. Math. Z. 18:1923;278-288.
31. Ore O. Zur Theorie der Eisensteinschen Gleichungen. Math. Z. 20:1924;267-279.
32. Ore O. Zur Theorie der Algebraisschen Körper. Acta Math. 44:1924;219-314.
33. Ostrowski A. M. On multiplication and factorization of polynomials, I. Lexicographic ordering and extreme aggregates of terms. Aequationes Math. 13:1975;201-228.
34. Ostrowski A. M. On multiplication and factorization of polynomials, II. Irreducibility discussion. Aequationes Math. 14:1976;1-32.
35. Rella T. Ordnungsbestimmungen in Integritätsbereichen und Newtonsche Polygone. J. Riene Angew. Math. 158:1927;33-48.
36. Schmidt W. M. Equations over Finite Fields: an Elementary Approach. Lecture Notes in Mathematics. 536:1976;Springer-Verlag, Berlin/New York.
37. Schneider R. Convex bodies: The Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications. 44:1993;Cambridge Univ. Press, Cambridge.
38. Schönemann T. Von denjenigen moduln, welche potenzen von primzahlen sind. J. Reine Angew. Math. 32:1846;93-105.
39. Shanok C. Convex polyhedra and criteria for irreducibility. Duke Math. J. 2:1936;103-111.
40. Shephard G. C. Decomposable convex polyhedra. Mathematika. 10:1963;89-95.
41. Smilansky Z. An indecomposable polytope all of whose facets are decomposable. Mathematika. 33:1986;192-196.
42. Smilansky Z. Decomposability of polytopes and polyhedra. Geom. Dedicata. 24:1987;29-49.
43. Stepanov S. A. Congruences with two unknowns. Izv. Akad. Nauk SSSR Ser. Mat. 36:1972;683-711.
44. Stepanov S. A. Rational points of algebraic curves over finite fields. Current Problems of Analytic Number Theory (Proc. Summer School, Minsk, 1972). 1974;Nauka, Minsk.
45. Stichtenoth H. Algebraic Function Fields and Codes. Universitext. 1993;Springer-Verlag, Berlin.
46. Sturmfels B. Gröbner Bases and Convex Polytopes. University Lecture Series. 8:1996;Am. Math. Soc. Providence.
47. Szönyi T. Some applications of algebraic curves infinite geometry and combinatorics. Railey R. A. Surveys in Combinatorics, 1997. London Mathematical Society Lecture Notes Series. 241:1997;Cambridge Univ. Press, Cambridge.
48. Wan D. Minimal polynomials and distinctness of Kloosterman sums. Finite Fields Appl. 1:1995;189-203.
49. Wan D. Newton polygons of zeta functions and L functions. Ann. of Math. (2). 137:1993;249-293.
50. Webster R. Convexity. 1994;Oxford Univ. Press, Oxford.
51. Williams K. S. Eisenstein's criteria for absolute irreducibility over a finite field. Canad. Math. Bull. 9:1966;575-580.
52. Ziegler G. M. Lectures on Polytopes. Graduate Texts in Mathematics. 152:1995;Springer-Verlag, Berlin/New York.