Waring Rank and Apolarity of Some Symmetric Polynomials

We examine lower bounds for the Waring rank for certain types of symmetric polynomials. The first are Schur polynomials, a symmetric polynomial indexed by integer partitions. We prove some results about the Waring rank of certain types of Schur polynomials, based on their integer partition. We also make some observations about the Waring rank in general for Schur polynomials, based on the shape of their Semistandard Young Tableaux. The second type of polynomials we refer to as a Power of a Fermat-type polynomial, or a PFT polynomial. This is a Fermat type (or power sum) polynomial over n variables with degree p taken to some power k. We prove this polynomial is not compressed when p > k and k > 2, and conjecture the result is true in general for all p. The proof takes the following form: the degree k + 1 annihilator ideal is examined and identified, and form of Rank-Nullity is applied, which provides a formula for the size of the degree k + 1 subspace of non-zero partial derivatives for that polynomial. Then we verify that this subspace is linearly independent, which gives us the dimension of the space of Derivs, and thus a lower bound for the Waring rank of that polynomial.