Root multiplicities and number of nonzero coefficients of a polynomial

Details Root multiplicities and number of nonzero coefficients of a polynomial Root multiplicities and number of nonzero coefficients of a polynomial

2005-12-12

arxiv.org/abs/math/0512239v2

J. Algebra Appl. 6 (2007), no. 3, 469--475

Mathematics

Number Theory

6 pages. Minor change from previous version: added Example 6, illustrating the difficulties arising when one tries to relax th

Scientific paper

10.1142/S0219498807002338

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate statement in positive characteristic. Furthermore, we present a new proof of the original result, which produces also the exact number of monic polynomials of a given degree for which the bound is attained. A similar argument allows us to determine the number of monic polynomials of a given degree, multiplicity of a given nonzero root, and number of nonzero coefficients, over a finite field of characteristic larger than the degree.

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