142 research outputs found
CHAMP: A Cherednik Algebra Magma Package
We present a computer algebra package based on Magma for performing
computations in rational Cherednik algebras at arbitrary parameters and in
Verma modules for restricted rational Cherednik algebras. Part of this package
is a new general Las Vegas algorithm for computing the head and the
constituents of a module with simple head in characteristic zero which we
develop here theoretically. This algorithm is very successful when applied to
Verma modules for restricted rational Cherednik algebras and it allows us to
answer several questions posed by Gordon in some specific cases. We could
determine the decomposition matrices of the Verma modules, the graded G-module
structure of the simple modules, and the Calogero-Moser families of the generic
restricted rational Cherednik algebra for around half of the exceptional
complex reflection groups. In this way we could also confirm Martino's
conjecture for several exceptional complex reflection groups.Comment: Final version to appear in LMS J. Comput. Math. 41 pages, 3 ancillary
files. CHAMP is available at http://thielul.github.io/CHAMP/. All results are
listed explicitly in the ancillary PDF document (currently 935 pages). Please
check the website for further update
Supersolvability of built lattices and Koszulness of generalized Chow rings
We give an explicit quadratic Grobner basis for generalized Chow rings of
supersolvable built lattices, with the help of the operadic structure on
geometric lattices introduced in a previous article. This shows that the
generalized Chow rings associated to minimal building sets of supersolvable
lattices are Koszul. As another consequence, we get that the cohomology
algebras of the components of the extended modular operad in genus 0 are
Koszul.Comment: Second version. Cleaned up a few proofs. Comments are welcom
The Complexity of Algebraic Algorithms for LWE
Arora & Ge introduced a noise-free polynomial system to compute the secret of
a Learning With Errors (LWE) instance via linearization. Albrecht et al. later
utilized the Arora-Ge polynomial model to study the complexity of Gr\"obner
basis computations on LWE polynomial systems under the assumption of
semi-regularity. In this paper we revisit the Arora-Ge polynomial and prove
that it satisfies a genericity condition recently introduced by Caminata &
Gorla, called being in generic coordinates. For polynomial systems in generic
coordinates one can always estimate the complexity of DRL Gr\"obner basis
computations in terms of the Castelnuovo-Mumford regularity and henceforth also
via the Macaulay bound.
Moreover, we generalize the Gr\"obner basis algorithm of Semaev & Tenti to
arbitrary polynomial systems with a finite degree of regularity. In particular,
existence of this algorithm yields another approach to estimate the complexity
of DRL Gr\"obner basis computations in terms of the degree of regularity. In
practice, the degree of regularity of LWE polynomial systems is not known,
though one can always estimate the lowest achievable degree of regularity.
Consequently, from a designer's worst case perspective this approach yields
sub-exponential complexity estimates for general, binary secret and binary
error LWE.
In recent works by Dachman-Soled et al. the hardness of LWE in the presence
of side information was analyzed. Utilizing their framework we discuss how
hints can be incorporated into LWE polynomial systems and how they affect the
complexity of Gr\"obner basis computations
Thomas Decomposition of Algebraic and Differential Systems
In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple
Generic interpolation polynomial for list decoding
AbstractWe extend results of K. Lee and M.E. OʼSullivan by showing how to use Gröbner bases to find the interpolation polynomial for list decoding a one-point AG code C=CL(rP,D) on any curve X, where P is an Fq-rational point on X and D=P1+P2+⋯+Pn is the sum of other Fq-rational points on X. We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial
Counting points on genus-3 hyperelliptic curves with explicit real multiplication
We propose a Las Vegas probabilistic algorithm to compute the zeta function
of a genus-3 hyperelliptic curve defined over a finite field ,
with explicit real multiplication by an order in a totally
real cubic field. Our main result states that this algorithm requires an
expected number of bit-operations, where the
constant in the depends on the ring and on
the degrees of polynomials representing the endomorphism . As a
proof-of-concept, we compute the zeta function of a curve defined over a 64-bit
prime field, with explicit real multiplication by .Comment: Proceedings of the ANTS-XIII conference (Thirteenth Algorithmic
Number Theory Symposium
Solving polynomial systems via symbolic-numeric reduction to geometric involutive form
AbstractWe briefly survey several existing methods for solving polynomial systems with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method
A verified Common Lisp implementation of Buchberger's algorithm in ACL2
In this article, we present the formal verification of a Common
Lisp implementation of Buchberger's algorithm for computing
Gröbner bases of polynomial ideals. This work is carried out in
ACL2, a system which provides an integrated environment where
programming (in a pure functional subset of Common Lisp) and
formal verification of programs, with the assistance of a theorem
prover, are possible. Our implementation is written in a real
programming language and it is directly executable within the
ACL2 system or any compliant Common Lisp system. We provide
here snippets of real verified code, discuss the formalization details
in depth, and present quantitative data about the proof effort
Efficiently and Effectively Recognizing Toricity of Steady State Varieties
We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gr\"obner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity or shifted toricity. In the positive case, these algorithms produce generating binomials. In addition, we propose an asymptotically fast algorithm for testing membership in a binomial variety over the algebraic closure of the rational numbers
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