Multiplicidade de anéis 1-dimensionais e uma aplicação ao problema de Waring

Abstract

Let k be an algebraically closed eld of characteristic zero and consider the polynomial ring S = k[x1, . . . , xn] endowed with the standard grading. The Waring's problem for a form F ∈ S of degree d asks about the least integer s ≥ 1 for which there exist linear forms L1, . . . , Ls ∈ S satisfying F = Σs i=1 Ldi. Such integer is called Waring rank of F. In this dissertation, we present a solution to this problem { due to Carlini-Catalisano-Geramita { in the case of monomials, as an interesting application of a theorem (due to the same authors) that establishes a lower bound for the multiplicity of (standard) graded, nitely generated, reduced, 1-dimensional k-algebras.