Abstract. We present new lower bounds for the symmetric border rank of the n × n determinant for all n. Further lower bounds are given for the 3 × 3 permanent.

The affine funtf variety is the Zariski closure of the set of finite unit norm tight frames. Determining the fiber of a projection of the funtf variety onto a set of coordinates is called the algebraic funtf completion problem. The algebraic matroid of an algebraic variety encodes the dimensions of fibers of coordinate projections. This work characterizes the bases of the algebraic matroid underlying the affine funtf variety of funtfs in R 3 , and partial results towards similar characterizations for funtfs in R n with n ≥ 4 are also given. We provide a method to bound the degree of the projections based of off combinatorial data.2010 Mathematics Subject Classification. 05B35.

We introduce homogenized funtf (finite tight unit norm frames) varieties and study the degrees of their coordinate projections. These varieties compactify the affine funtf variety differently from the projectivizations studied in [12]. However, each are the closures (Zariski) of the set of finite tight unit norm frames. Our motivation comes from studying the algebraic frame completion problem.