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- W2891175669 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 $$(d+1)^{n+2}$$ tabulated values of P to produce the value of P at any of the $$q^n$$ points using $$O(nqd^2)$$ 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 $$(d/s+1)^{n+s}$$ tabulated values to produce the value of P at any point using $$Oleft( nq^ssqright) $$ 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 $$mtimes m$$ integer matrix with entries bounded in absolute value by a constant can be computed in time $$2^{m-Omega left( sqrt{m/log log m}right) }$$ , improving an earlier algorithm of Björklund (in: Proceedings of the 15th SWAT, vol 17, pp 1–11, 2016) that runs in time $$2^{m-Omega left( sqrt{m/log m}right) }$$ ." @default.
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- W2891175669 date "2018-09-18" @default.
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- W2891175669 title "Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants" @default.
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- W2891175669 doi "https://doi.org/10.1007/s00453-018-0513-7" @default.
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