38,936 research outputs found
On reconstructing subvarieties from their periods
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
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
On asymptotically good ramp secret sharing schemes
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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