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Nouvelle majoration de la norme des facteurs d’un polynĂ´me. (New upper bounds for the norm of the factors of a polynomial). (French) Zbl 0729.12001
The author proves a number of inequalities concerning the factorization of a polynomial F with complex coefficients into a product \(F=PQ\). In the following, \(| P| =\max \{| P(z)|:| z| =1\}\), M(P) denotes Mahler’s measure of P (the geometric mean of \(| P(z)|\) on the unit circle), and the degrees of P and Q are p and q, respectively. His main result is that \[ | P| M(Q)\leq 2^{- q}\frac{(p+q)^{p+q}}{p^ pq^ q}| PQ|. \] As a corollary, he obtains the following result: suppose that F is of degree n and that P is a factor of F for which the leading coefficient is at most as large as the leading coefficient of F, then \(| P| \leq (3/2)^ n | F|\). This improves a result of A. Granville [Monatsh. Math. 109, 271-277 (1990; Zbl 0713.12001)], who obained the same result with 3/2 replaced by the golden ratio (\(\sqrt{5}+1)/2\).

12D05 Polynomials in real and complex fields: factorization
26D05 Inequalities for trigonometric functions and polynomials
30C10 Polynomials and rational functions of one complex variable