Generalized Kakeya Sets for Polynomial Evaluation and Faster Computation of Fermionants

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)n+2 tabulated values of P to produce the value of P at any of the q points using O(nqd2) 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)n+s tabulated values to produce the value of P at any point using O(nqsq) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (2004), Saraf and Sudan (2008), and Dvir (2009) 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 (2011) that captures numerous fundamental algebraic and combinatorial invariants 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 (√ m/ log logm ) , improving an earlier algorithm of Björklund (2016) that runs in time 2 (√ m/ logm ) . 1998 ACM Subject Classification F.2.1, F.2.2, G.2.1, G.2.2