The monic rank and instances of Shapiro's Conjecture

Shapiro's Conjecture states that every homogeneous polynomial in C[x,y] of degree de is the sum of at most d d-th powers of homogeneous polynomials of degree e. In this talk, we prove a few instances of this conjecture. We do this by introducing the monic rank. For a polynomial, the monic rank is the minimal k for which we can write (an appropriately scaled version of) the polynomial as a sum of k d-th powers of monic polynomials. We prove that the maximal monic rank of polynomials of degree de is at most d when e=1, d=1,2 or (d,e)=(3,2),(3,3),(3,4),(4,2). This is joint work with Jan Draisma, Alessandro Oneto and Emanuele Ventura.