Title: A tale of two ranks
Abstract: Let k be a field and Q in k[x_1,...,x_n] a form (homogeneous polynomial) of degree d>1. Here are two ways to measure the degeneracy of Q:
- Algebraic: Schmidt rank is the smallest natural number r such that Q = S_1T_1+...S_rT_r with S_i,T_i in k[x_1,...,x_n] forms of positive degree. It was introduced by Schmidt in 1985.
- Geometric: Birch rank is the codimension in affine n-space of the variety where the gradient of Q vanishes. This was introduced by Birch in 1962.
Both Birch and Schmidt defined rank when Q has rational coefficients and showed that if Q is of large rank then a local-to-global formula holds for counting integer points on the variety Q=0. Birch's result has subsequently been extended to give local-to-global formulas in many other settings. I will discuss a recent result showing that Schmidt rank and Birch rank are closely related for many fields (among them number fields, finite fields, and function fields), and a corollary about counting prime points on the variety Q=0.
Joint work with Tamar Ziegler.