Zero-free angular sectors and lens-shaped regions for polynomials, with applications to irreducibility

References

• Ballieu, R. (1947). Sur les limitations des racines d’une équation algébrique. Acad. Roy. Belg. Bull. Cl. Sci. 33(5):747–750.
   Google Scholar
• Batra, P., Mignotte, M., Ştefănescu, D. (2017). Improvements of Lagrange’s bound for polynomial roots. J. Symbolic Comput. 82:19–25. DOI: 10.1016/j.jsc.2016.10.001.
   Web of Science ®Google Scholar
• Bodin, A., Dèbes, P., Najib, S. (2020). Prime and coprime values of polynomials. Enseign. Math. 66(1):173–182. DOI: 10.4171/LEM/66-1/2-9.
   Google Scholar
• Bonciocat, A.I., Bonciocat, N.C. (2009). The irreducibility of polynomials that have one large coefficient and take a prime value. Can. math. Bull. 52(4):511–520. DOI: 10.4153/CMB-2009-052-9.
   Web of Science ®Google Scholar
• Bonciocat, A.I., Bonciocat, N.C., Zaharescu, A. (2011). On the irreducibility of polynomials that take a prime power value. Bull. Math. Soc. Sci. Math. Roumanie. 54(102): 41–54.
   Google Scholar
• Brillhart, J., Filaseta, M., Odlyzko, A. (1981). On an irreducibility theorem of A. Cohn. Can. J. Math. 33(5):1055–1059. DOI: 10.4153/CJM-1981-080-0.
   Web of Science ®Google Scholar
• Cole, M., Dunn, S., Filaseta, M. (2016). Further irreducibility criteria for polynomials with non-negative coefficients. Acta Arith. 175(2):137–181. DOI: 10.4064/aa8376-5-2016.
   Web of Science ®Google Scholar
• Cowling, V.F., Thron, W.J. (1954). Zero-free regions of polynomials. Amer. Math. Month. 61(10):682–687. DOI: 10.1080/00029890.1954.11988549.
   Google Scholar
• Cowling, V.F., Thron, W.J. (1956). Zero-free regions of polynomials. J. Indian Math. Soc. (N.S.) 20:307–310.
   Google Scholar
• Dorwart, H.L. (1935). Irreducibility of polynomials. Amer. Math. Monthly 42(6):369–381. DOI: 10.1080/00029890.1935.11987732.
   Google Scholar
• Filaseta, M. (1982). A further generalization of an irreducibility theorem of A. Cohn. Canad. J. Math. 34(6):1390–1395.
   Web of Science ®Google Scholar
• Filaseta, M. (1988). Irreducibility criteria for polynomials with non-negative coefficients. Can. J. Math. 40(2):339–351. DOI: 10.4153/CJM-1988-013-6.
   Web of Science ®Google Scholar
• Fujiwara, M. (1916). Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung. Tôhoku Math. J. 10:167–171.
   Google Scholar
• Girstmair, K. (2005). On an Irreducibility Criterion of M. Ram Murty. Amer. Math. Monthly 112(3):269–270. DOI: 10.2307/30037445.
   Web of Science ®Google Scholar
• Guersenzvaig, N.H. (2013). Simple arithmetical criteria for irreducibility of polynomials with integer coefficients. Integers 13:1–21.
   Google Scholar
• Guersenzvaig, N.H., Szechtman, F. (2012). Roots multiplicity and square free factorization of polynomials using companion matrices. Linear Algebra Appl. 436(9):3160–3164. DOI: 10.1016/j.laa.2011.10.018.
   Web of Science ®Google Scholar
• Kempner, A.J. (1922). Ueber die Separation komplexer Wurzeln algebraischer Gleichungen. Math. Ann. 85(1):49–59. DOI: 10.1007/BF01449601.
   Google Scholar
• Kioustelidis, J.B. (1986). Bounds for positive roots of polynomials. J. Comput. Appl. Math. 16(2):241–244. DOI: 10.1016/0377-0427(86)90096-8.
   Web of Science ®Google Scholar
• Kojima, T. (1914). On a theorem of Hadamard’s and its application. Tôhoku Math. J. 5:54–60.
   Google Scholar
• László, L. (1982). Imaginary Part Bounds on Polynomial Zeros. Linear Algebra Appl. 44:173–180. DOI: 10.1016/0024-3795(82)90013-1.
   Web of Science ®Google Scholar
• Marden, M. (1966). Geometry of Polynomials, Mathematical Surveys and Monographs No. 3. Providence, RI: American Mathematical Society.
   Google Scholar
• Mignotte, M., Ştefănescu, D. (1999). Polynomials. An Algorithmic Approach. Singapore: Springer.
   Google Scholar
• Ore, O. (1934). Einige Bemerkungen über Irreduzibilität. Jahresbericht der Deutschen Mathematiker-Vereinigung. 44:147–151.
   Google Scholar
• Perron, O. (1951). Algebra. II Theorie der algebraischen Gleichungen. Berlin: Walter de Gruyter & Co.
   Google Scholar
• Pólya, G., Szegö, G. (1964). Aufgaben und Lehrsätze aus der Analysis. Berlin: Springer-Verlag.
   Google Scholar
• Murty, M. R. (2002). Prime numbers and irreducible polynomials. Amer. Math. Monthly. 109(5):452–458. DOI: 10.1080/00029890.2002.11919872.
   Web of Science ®Google Scholar
• Stäckel, P. (1918). Arithmetischen Eigenschaften ganzer Funktionen. J. Mathematik. 148:101–112.
   Google Scholar
• Ştefănescu, D. (2013). On bounds for real roots of polynomials. Rom. J. Phys. 58(9–10):1428–1435. nos.
   Web of Science ®Google Scholar
• Turán, P. (1954). Hermite-expansion and strips for zeros of polynomials. Arch. Math. 5(1-3):148–152. DOI: 10.1007/BF01899331.
   Google Scholar
• Weisner, L. (1934). Criteria for the irreducibility of polynomials. Bull. Amer. Math. Soc. 40(12):864–870. DOI: 10.1090/S0002-9904-1934-05989-5.
   Google Scholar