Monomials as Sums of k th-Powers of Forms

Abstract

Motivated by recent results on the Waring problem for polynomial rings [4] and representation of monomial as sum of powers of linear forms [3], 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.

Key Words:

• Monomials
• Sum of powers
• Waring problems

2010 Mathematics Subject Classification:

• 13P05
• 11P05
• 13F20

ACKNOWLEDGMENTS

The authors wish to thank Froeberg and Shapiro for their many helpful suggestions and wonderful ideas.

Notes

Communicated by R. Piene.