On generic and maximal k-ranks of binary forms

Binary Forms with Three Different Relative Ranks

Suppose f(x, y) is a binary form of degree d with coefficients in a field K ⊆ C. The K-rank of f is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d ≥ 5, there always exists a form of degree d with at least three different ranks over various fields. The K-rank of a form f (such as x3y2) may depend on whether -1 is a sum of two s...

On Maximum, Typical, and Generic Ranks

We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the maximum value of the real rank is at most twice the smallest typical rank, which is equal to the (complex) generic rank.

Recent results on p - ranks and Smith normal forms of some 2 − ( v , k , λ ) designs

We survey recent results on p-ranks and Smith normal forms of some 2 − (v, k, λ) designs. In particular, we give a description of the recent work in [11] on the Smith normal forms of the 2-designs arising from projective and affine spaces over Fq .

Dyson’s Ranks and Maass Forms

which were defined by Ramanujan and Watson decades ago. In his last letter to Hardy dated January 1920 (see pages 127-131 of [27]), Ramanujan lists 17 such functions, and he gives 2 more in his “Lost Notebook” [27]. In his paper “The final problem: An account of the mock theta functions” [32], Watson defines 3 further functions. Surprisingly, much remains unknown about these enigmatic series. R...

On the Maximal Degree of the K-Step Obrechkoff’s Method