# Erik Jernqvist: Waring decompositions

**Time: **
Wed 2019-01-30 14.30 - 15.30

**Location: ** Room 32, House 5, Kräftriket, Department of Mathematics, Stockholm University

**Respondent: **
Erik Jernqvist

**Supervisor: **
Samuel Lundqvist

Abstract: *A Waring decomposition of a homogenous polynomial f is a sum of powers of linear forms expressing f The Waring rank is the smallest possible number of summands in such a decomposition. In this thesis we focus on decompositions of monomials. For decomposistions of monomials over the complex numbers the Waring rank is known. For decompositions over the reals or the rationals the Waring rank is not known, apart from two classes of monomials. For monomials with the smallest exponent equal to one and two-variable monomials, the Waring rank over the rationals and the reals are known and coincides. For these cases we give explicit constructions that yield minimal decompositions. For other monomials we examine upper bounds for the ranks over the rationals and the reals. We also present some results for polynomials.*