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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. 

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. In spite of the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas. We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, 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 size of set-multi-linearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept we rederive known derandomization results based on the generator by Shpilka and Volkovich, and present a new application for read-once oblivious arithmetic branching programs that provably transcends the usual combinatorial techniques. 

Constraining contacts to remain fixed on an object during manipulation limits the potential workspace size, as motion is subject to the hand’s kinematic topology. Finger gaiting is one way to alleviate such restraints. It allows contacts to be freely broken and remade so as to operate on different manipulation manifolds. This capability, however, has traditionally been difficult or impossible to practically realize. A finger gaiting system must simultaneously plan for and control forces on the object while maintaining stability during contact switching. This letter alleviates the traditional requirement by taking advantage of system compliance, allowing the hand to more easily switch contacts while maintaining a stable grasp. Our method achieves complete SO(3) finger gaiting control of grasped objects against gravity by developing a manipulation planner that operates via orthogonal safe modes of a compliant, underactuated hand absent of tactile sensors or joint encoders. During manipulation, a low-latency 6D pose object tracker provides feedback via vision, allowing the planner to update its plan online so as to adaptively recover from trajectory deviations. The efficacy of this method is showcased by manipulating both convex and non-convex objects on a real robot. Its robustness is evaluated via perturbation rejection and long trajectory goals. To the best of the authors’ knowledge, this is the first work that has autonomously achieved full SO(3) control of objects within-hand via finger gaiting and without a support surface, elucidating a valuable step towards realizing true robot in-hand manipulation capabilities.