Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

Abstract

We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q − 1, our first data structure relies on (d + 1)[superscript n+2] tabulated values of P to produce the value of P at any of the q[superscript n] points using O(nqd[superscript 2]) arithmetic operations in the finite field. Assuming that s divides d and d/s divides q − 1, our second data structure assumes that P satisfies a degree-separability condition and relies on (d/s + 1)[superscript n+s] tabulated values to produce the value of P at any point using O(nq[superscript s]sq) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (Duke Math J 121(1):35–74, 2004), Saraf and Sudan (Anal PDE 1(3):375–379, 2008) and Dvir (Incidence theorems and their applications, 2012. arXiv:1208.5073) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for computing integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (Partition functions of strongly correlated electron systems as fermionants, 2011. arXiv:1108.2461v1) that captures numerous fundamental algebraic and combinatorial functions such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics. In particular, a corollary of our main theorem for fermionants is that the permanent of an m × m integer matrix with entries bounded in absolute value by a constant can be computed in time 2[superscript m−(√m/ log log m)], improving an earlier algorithm of Björklund (in: Proceedings of the 15th SWAT, vol 17, pp 1–11, 2016) that runs in time 2[superscript m−(√m/ log m)].