The Solution to Waring's Problem for Monomials

Abstract

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.

References in corpus (1)

• On the variety parametrizing completely decomposable polynomials

Cited by in corpus (8)

• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• On the cactus rank of cubics forms
• Waring decompositions of monomials
• Border Ranks of Monomials
• Typical Real Ranks of Binary Forms
• A comparison of different notions of ranks of symmetric tensors
• The Waring rank of the sum of pairwise coprime monomials
• Positive Gorenstein Ideals