{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10313","name":"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)^{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.\r\n\r\nAs 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"],"author":[{"@type":"Person","name":"Bj\u00f6rklund, Andreas","givenName":"Andreas","familyName":"Bj\u00f6rklund"},{"@type":"Person","name":"Kaski, Petteri","givenName":"Petteri","familyName":"Kaski"},{"@type":"Person","name":"Williams, Ryan","givenName":"Ryan","familyName":"Williams"}],"position":6,"pageStart":"6:1","pageEnd":"6:13","dateCreated":"2018-03-02","datePublished":"2018-03-02","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bj\u00f6rklund, Andreas","givenName":"Andreas","familyName":"Bj\u00f6rklund"},{"@type":"Person","name":"Kaski, Petteri","givenName":"Petteri","familyName":"Kaski"},{"@type":"Person","name":"Williams, Ryan","givenName":"Ryan","familyName":"Williams"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.IPEC.2017.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1016\/S0020-0190(96)00159-7","http:\/\/dx.doi.org\/10.1007\/BF01171101","http:\/\/dx.doi.org\/10.4230\/LIPIcs.SWAT.2016.17","http:\/\/dx.doi.org\/10.1016\/j.ipl.2017.04.015","http:\/\/dx.doi.org\/10.1145\/2933057.2933101","http:\/\/dx.doi.org\/10.4230\/LIPIcs.ICALP.2017.91","http:\/\/arxiv.org\/abs\/1108.2461v1","http:\/\/dx.doi.org\/10.1007\/978-3-642-39206-1_31","http:\/\/eccc.hpi-web.de\/report\/2009\/077","http:\/\/arxiv.org\/abs\/1208.5073","http:\/\/dx.doi.org\/10.1137\/08073408X","http:\/\/dx.doi.org\/10.1007\/s10801-011-0274-8","http:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i3p36","http:\/\/dx.doi.org\/10.1145\/146585.146605","http:\/\/arxiv.org\/abs\/1110.1821","http:\/\/dx.doi.org\/10.1215\/S0012-7094-04-12112-8","http:\/\/dx.doi.org\/10.2140\/apde.2008.1.375","http:\/\/dx.doi.org\/10.1145\/800135.804419","http:\/\/dx.doi.org\/10.1016\/0304-3975(79)90044-6","http:\/\/dx.doi.org\/10.1017\/CBO9781139856065","http:\/\/dx.doi.org\/10.4230\/LIPIcs.CCC.2016.2"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6292","volumeNumber":89,"name":"12th International Symposium on Parameterized and Exact Computation (IPEC 2017)","dateCreated":"2018-03-02","datePublished":"2018-03-02","editor":[{"@type":"Person","name":"Lokshtanov, Daniel","givenName":"Daniel","familyName":"Lokshtanov"},{"@type":"Person","name":"Nishimura, Naomi","givenName":"Naomi","familyName":"Nishimura"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10313","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6292"}}}