Jean Cardinal


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,…,ak from the set such that F(a1,…,ak)=0. We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n 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 improvements over the naive O*(n^{k-1})
algorithm for this problem (where the O*(.)
notation omits subpolynomial factors), in both the real RAM and algebraic decision tree models of computation.

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.

This is joint work with Micha Sharir.
https://arxiv.org/abs/2212.03030