Monomials as Sums ofk th-Powers of Forms
Enrico Carlini, Alessandro Oneto · Communications in Algebra · 2014
Motivated by recent results on the Waring problem for polynomial rings [4 Fröberg , R. , Ottaviani , G. , Shapiro , B. ( 2012 ). On the waring problem for polynomial rings . Proceedings of the National Academy of Sciences 109 ( 15 ): 5600 – 5602 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] and representation of monomial as sum of powers of linear forms [3 Carlini , E. , Catalisano , M. V. , Geramita , A. V. ( 2012 ). The solution to the waring problem for monomials and the sum of coprime monomials . Journal of Algebra 370 : 5 – 14 .[Crossref], [Web of Science ®] , [Google Scholar]], we consider the problem of presenting monomials of degree kd as sums of kth-powers of forms of degree d. We produce a general bound on the number of summands for any number of variables which we refine in the two variables case. We completely solve the k = 3 case for monomials in two and three variables.