## Upper bound for the norm of the factors of a polynomial: case where all the roots of the polynomial are real. (Majoration de la norme des facteurs d’un polynôme: Cas où toutes les racines du polynôme sont réelles.)(French)Zbl 0778.12001

Author’s abstract: In the first part of this paper, we give upper bounds for the norm of factors of polynomials whose zeros are real. These bounds are very useful in polynomial factorization algorithms. In the second part, we prove an upper bound for the number of real roots of any polynomial, and also a lower bound for the norm of a polynomial, depending on the distribution of its roots in the complex plane.

### MSC:

 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 12D05 Polynomials in real and complex fields: factorization 26C10 Real polynomials: location of zeros 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
Full Text:

### References:

 [1] 1. B. BEAUZAMY, Degree-free upper estimates in polynomial factorizations, (manuscrit). [2] 2. P. ERDÖS, P. TURÁN, On the distribution of the roots of polynomials, Ann. Math., 51, 1950, p. 105-119. Zbl0036.01501 MR33372 · Zbl 0036.01501 [3] 3. T. GANELIUS, Sequences of analytic functions and their zeros, Arkiv för Math, 3, 1954-1958, p. 1-50. Zbl0055.06905 MR62826 · Zbl 0055.06905 [4] 4. Ph. GLESSER, Bornes pour les algorithmes de factorisation des polynômes, Thèse de Doctorat. [5] 5. Ph. GLESSER, Nouvelle majoration de la norme des facteurs d’un polynôme, Comptes-Rendus de l’Acad. Roy. du Canada, XII, n^\circ 6, 1990, p. 224-228. Zbl0729.12001 MR1088308 · Zbl 0729.12001 [6] 6. D. E. KNUTH, The art of computer programming, Vol. 2, Seminumerical algorithms, Addison-Wesley, 1981. MR633878 · Zbl 0477.65002 [7] 7. E. LANDAU, Sur quelques théorèmes de M. Petrovic relatifs aux zéros des fonctions analytiques, Bull. Soc. Math, de France., 33, 1905, p. 251-261. MR1504527 JFM36.0467.01 · JFM 36.0467.01 [8] 8. A. K. LENSTRA, H. W. Jr. LENSTRA, L. LOVÀSZ, Factoring polynomials with rational coefficients, Math. Ann., 261, 1982, p. 515-534. Zbl0488.12001 MR682664 · Zbl 0488.12001 [9] 9. M. MIGNOTTE, An inequality about factors of polynomials, Math. Comp., 28, 1974, p. 1153-1157. Zbl0299.12101 MR354624 · Zbl 0299.12101 [10] 10. M. MIGNOTTE, An inequality about irreducible factors of integer polynomials, J. of Number Theory, 30, 1988, p. 156-166. Zbl0648.12002 MR961913 · Zbl 0648.12002 [11] 11. M. MIGNOTTE, Ph. GLESSER, An inequality about irreducible factors of integer polynomials (II), SYMSAC August 1990 (Tokyo). MR1123956 · Zbl 0739.11010 [12] 12. I. SCHUR, Preuss. Akad. Wiss. Sitzungsber. 1933, p. 403-428. Zbl0007.00101 · Zbl 0007.00101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.