We initiate a systematic analytic study of the power of hitting set generators by characterizing their vanishing ideals, i.e., the sets of polynomials that they fail to hit. We provide two such characterizations for our generator. First, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition class size of set-multilinearity in the vanishing ideal. Second, inspired by a connection to alternating algebra, we develop a structured deterministic membership test for the multilinear part of the vanishing ideal. We present a derivation based on alternating algebra along with the required background, as well as one in terms of zero substitutions and partial derivatives, avoiding the need for alternating algebra.
As evidence of the utility of our analytic approach, we rederive known derandomization results based on the generator by Shpilka and Volkovich and present a new application in derandomization / lower bounds for read-once oblivious algebraic branching programs.