158 research outputs found
Computing topological zeta functions of groups, algebras, and modules, II
Building on our previous work (arXiv:1405.5711), we develop the first
practical algorithm for computing topological zeta functions of nilpotent
groups, non-associative algebras, and modules. While we previously depended
upon non-degeneracy assumptions, the theory developed here allows us to
overcome these restrictions in various interesting cases.Comment: 33 pages; sequel to arXiv:1405.571
Signature Gr\"obner bases in free algebras over rings
We generalize signature Gr\"obner bases, previously studied in the free
algebra over a field or polynomial rings over a ring, to ideals in the mixed
algebra where is a principal
ideal domain. We give an algorithm for computing them, combining elements from
the theory of commutative and noncommutative (signature) Gr\"obner bases, and
prove its correctness.
Applications include extensions of the free algebra with commutative
variables, e.g., for homogenization purposes or for performing ideal theoretic
operations such as intersections, and computations over as
universal proofs over fields of arbitrary characteristic.
By extending the signature cover criterion to our setting, our algorithm also
lifts some technical restrictions from previous noncommutative signature-based
algorithms, now allowing, e.g., elimination orderings. We provide a prototype
implementation for the case when is a field, and show that our algorithm
for the mixed algebra is more efficient than classical approaches using
existing algorithms.Comment: 10 page
Algorithmic Boundedness-From-Below Conditions for Generic Scalar Potentials
Checking that a scalar potential is bounded from below (BFB) is an ubiquitous
and notoriously difficult task in many models with extended scalar sectors.
Exact analytic BFB conditions are known only in simple cases. In this work, we
present a novel approach to algorithmically establish the BFB conditions for
any polynomial scalar potential. The method relies on elements of multivariate
algebra, in particular, on resultants and on the spectral theory of tensors,
which is being developed by the mathematical community. We give first a
pedagogical introduction to this approach, illustrate it with elementary
examples, and then present the working Mathematica implementation publicly
available at GitHub. Due to the rapidly increasing complexity of the problem,
we have not yet produced ready-to-use analytical BFB conditions for new
multi-scalar cases. But we are confident that the present implementation can be
dramatically improved and may eventually lead to such results.Comment: 27 pages, 2 figures; v2: added reference
Algorithms for graded injective resolutions and local cohomology over semigroup rings
AbstractLet Q be an affine semigroup generating Zd, and fix a finitely generated Zd-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Zd-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules HIi(M) supported on any monomial (that is, Zd-graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them
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