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Generalized Kakeya Sets for Polynomial Evaluation and Faster Computation of Fermionants

Authors Andreas Björklund, Petteri Kaski, Ryan Williams



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Andreas Björklund
Petteri Kaski
Ryan Williams

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Andreas Björklund, Petteri Kaski, and Ryan Williams. Generalized Kakeya Sets for Polynomial Evaluation and Faster Computation of Fermionants. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 6:1-6:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.IPEC.2017.6

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 O(nq^ssq) 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-by-m integer matrix with entries bounded in absolute value by a constant can be computed in time 2^{m-Omega(sqrt(m/log log m))}, improving an earlier algorithm of Bjorklund (2016) that runs in time 2^{m-Omega(sqrt(m/log m))}.
Keywords
  • Besicovitch set
  • fermionant
  • finite field
  • finite vector space
  • Hamiltonian cycle
  • homogeneous polynomial
  • Kakeya set
  • permanent
  • polynomial evaluatio

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