The Solution to Waring's Problem for Monomials

2011/10/04 by Enrico Carlini, Carlini, Enrico, Maria Virginia Catalisano +3
Mathematics · #13P05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13P05

paper · pdf · doi:10.48550/arxiv.1110.0745

arxiv created 2011/10/04 · arxiv updated 2011/10/05

Abstract

In the polynomial ring T=k[y₁,...,y_n], with n>1, we bound the multiplicity of homogeneous radical ideals I⊂ (y₁^a₁,...,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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