38,936 research outputs found

    On reconstructing subvarieties from their periods

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    We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then verifies the results. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. A highlight of the method is that the Picard group computations are proved to be correct despite the fact that the Picard numbers of our examples are not extremal.Comment: 16 pages; computational aspects highlighted; reconstruction of twisted cubics in higher degree surfaces adde

    Reconstructing Rational Functions with FireFly\texttt{FireFly}

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    We present the open-source C++\texttt{C++} library FireFly\texttt{FireFly} for the reconstruction of multivariate rational functions over finite fields. We discuss the involved algorithms and their implementation. As an application, we use FireFly\texttt{FireFly} in the context of integration-by-parts reductions and compare runtime and memory consumption to a fully algebraic approach with the program Kira\texttt{Kira}.Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio

    On asymptotically good ramp secret sharing schemes

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    Asymptotically good sequences of linear ramp secret sharing schemes have been intensively studied by Cramer et al. in terms of sequences of pairs of nested algebraic geometric codes. In those works the focus is on full privacy and full reconstruction. In this paper we analyze additional parameters describing the asymptotic behavior of partial information leakage and possibly also partial reconstruction giving a more complete picture of the access structure for sequences of linear ramp secret sharing schemes. Our study involves a detailed treatment of the (relative) generalized Hamming weights of the considered codes
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