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It is known that the weight (that is, the number of nonzero coefficients) ofa univariate polynomial over a field of characteristic zero is larger than themultiplicity of any of its nonzero roots. We extend this result to anappropriate statement in positive characteristic. Furthermore, we present a newproof of the original result, which produces also the exact number of monicpolynomials of a given degree for which the bound is attained. A similarargument allows us to determine the number of monic polynomials of a givendegree, multiplicity of a given nonzero root, and number of nonzerocoefficients, over a finite field of characteristic larger than the degree.Comment: 6 pages. Minor change from previous version: added Example 6,  illustrating the difficulties arising when one tries to relax the hypothesis  n<p of Theorem

ROOT MULTIPLICITIES AND NUMBER OF NONZERO COEFFICIENTS OF A POLYNOMIAL