Improved Algebraic Degeneracy Testing

In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers a₁,…,a_k from the set such that F(a₁,…,a_k)=0. We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve O^∗(n^k-1) algorithm for this problem (where the O^∗(·) notation omits subpolynomial factors). We show that the problem can be solved in time O^∗n^k-2+4k+2 for even k and in time O^∗n^k-2+4k-8k2-5 for odd k in the real RAM model of computation. We also prove that for k=4, the problem can be solved in time O^∗(n^2.625) in the algebraic decision tree model, and for k=5 it can be solved in time O^∗(n^3.56) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.