The Solution to Waring's Problem for Monomials

Details The Solution to Waring's Problem for Monomials The Solution to Waring's Problem for Monomials

2011-10-04

arxiv.org/abs/1110.0745v1

Mathematics

Commutative Algebra

Scientific paper

In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve the Waring Problem for all monomials, i.e. we compute the minimal number of linear forms needed to write a monomial as a sum of powers of these linear forms. Moreover, we give an explicit description of a sum of powers decomposition for monomials. We also produce new bounds for the Waring rank of polynomials which are a sum of pairwise coprime monomials.

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