Products of polynomials and a priori estimates for coefficients in polynomial decompositions: A sharp result

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Using a result of E. Bombieri which appeared in Beauzamy, Bombieri, Enflo and Montgomery (1990), we introduce a weighted norm on the space of polynomials in one variable ; the weights are much smaller than 1. For this norm, we give a precise estimate for products of polynomials : the numerical constant we obtain is best possible. The norm, moreover, is submultiplicative.

If P is a polynomial with integer coefficients, we deduce from the above result an a priori estimate for the size of the coefficients in any factor of P, which strongly improves all the existing ones.

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