An Algorithm to Compute the H-Bases for Ideals of Subalgebras

The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgebras, termed as SH-bases. In this paper, we present an analogue of H-bases for finitely generated ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.” We present their connection to the SAGBI-Gröbner basis concept, characterize HSG-basis, and show how to construct them.

Examples of computation of level lines of polynomials in a plane

Here we present a theory and 3 nontrivial examples of level lines calculating of real polynomials in the real plane. For this case we implement the following programs of computational algebra: factorization of a polynomial, calculation of the Grobner basis, construction of Newton's polygon, representation of an algebraic curve in a plane. Furthermore, it is shown how to overcome computational difficulties.

Computing with knot quandles

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

Improved rank bounds from 2-descent on hyperelliptic Jacobians

We describe a qualitative improvement to the algorithms for performing [Formula: see text]-descents to obtain information regarding the Mordell–Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra System and as a result, the rank bounds for hyperelliptic Jacobians are now sharper and have the conjectured parity.