Bruce Reznick: On the relative Waring rank of a binary form

Abstract: Suppose p is a binary form of degree d with coefficients in a number field F. The F-rank of p is the shortest way to write p as a linear combination of d-th powers of linear form, with coefficients in F. Back in the 1850s, J J Sylvester gave an algorithm for determining the C-rank of any binary form, and it turns out this algorithm also works over F.
(The question is much harder for forms in several variables and is called the symmetric tensor rank.) But it is possible that the rank can depend on F. For example, \((x + i y)^d + (x - i y)^d\) is a real form, with obvious C-rank 2. However, it can be shown that its R-rank equals d. My recent PhD student Neriman Tokcan and I showed that if d is at least 5, then forms can have three different ranks; basic examples are \((xy)^k\) and \((xy)^k(x+y)\). Nobody knows about 4. All methods of proof will be elementary.