Lower bounds for linear degeneracy testing

In the late nineties, Erickson proved a remarkable lower bound on the decision tree complexity of one of the central problems of computational geometry: given n numbers, do any r of them add up to 0? His lower bound of Ω(n ^r/2), for any fixed r, is optimal if the polynomials at the nodes are linear and at most r-variate. We generalize his bound to s-variate polynomials for s > r. Erickson's bound decays quickly as r grows and never reaches above pseudo-polynomial: we provide an exponential improvement. Our arguments are based on three ideas: (i) a geometrization of Erickson's proof technique; (ii) the use of error-correcting codes; and (iii) a tensor product construction for permutation matrices.