  We introduce a hitting set generator for Polynomial Identity Testing based on
evaluations of low-degree univariate rational functions at abscissas associated
with the variables. We establish an equivalence up to rescaling with a
generator introduced by Shpilka and Volkovich, which has a similar structure
but uses multivariate polynomials.
  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.
