## 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 ($$\sqrt{5}+1)/2$$.

### MSC:

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

Zbl 0713.12001