On Vinberg ( − 1,1) rings

We give a description of a 2-torsion free Vinberg ([Formula: see text]) ring [Formula: see text]. If every nonzero root space of [Formula: see text] for [Formula: see text] is one-dimensional where [Formula: see text] is a split abelian Cartan subring of [Formula: see text] which is nil on [Formula: see text] then [Formula: see text] is a Lie ring isomorphic to [Formula: see text]. This generalizes the known result obtained by Myung for the case that [Formula: see text] is a 2-torsion free Vinberg ([Formula: see text]) ring and is power associative. We also give a condition that a Levi factor [Formula: see text] of [Formula: see text] be an ideal of [Formula: see text] when the solvable radical of [Formula: see text] is nilpotent. We apply these results for reductive case of [Formula: see text].

ROOT MULTIPLICITIES AND NUMBER OF NONZERO COEFFICIENTS OF A POLYNOMIAL

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.