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 \(a_1,\ldots ,a_k\) from the set such that \(F(a_1,\ldots ,a_k) = 0\). 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 \(O^*(\cdot )\) notation omits subpolynomial factors). We show that the problem can be solved in time \(O^*\left( n^{k - 2 + \frac{4}{k+2}}\right) \) for even k and in time \(O^*\left( n^{k - 2 + \frac{4k-8}{k^2-5}}\right) \) 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.