Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.Comment: 8 page

HPC-GAP: engineering a 21st-century high-performance computer algebra system

Symbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multi-core to high-performance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes thatdo not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data. This paper describes a new implementation of the free open-source GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and cross-platform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory multi-core nodes, mid-scale distributed clusters of (multi-core) nodes, and full-blown HPC systems, comprising large-scale tightly-connected networks of multi-core nodes. This requires us to develop new cross-layer programming abstractions in the form of new domain-specific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on high-performance systems comprising up to 32,000 cores, as well as on ubiquitous multi-core systems and distributed clusters. The work reported here paves the way towards full scale exploitation of symbolic computation by high-performance computing systems, and we demonstrate the potential with two major case studies

On the complexity of computing Gr\"obner bases for weighted homogeneous systems

Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights $W=(w\_{1},\dots,w\_{n})$, $W$-homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree $\deg\_{W}(X\_{1}^{\alpha\_{1}},\dots,X\_{n}^{\alpha\_{n}}) = \sum w\_{i}\alpha\_{i}$. Gr\"obner bases for weighted homogeneous systems can be computed by adapting existing algorithms for homogeneous systems to the weighted homogeneous case. We show that in this case, the complexity estimate for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be divided by a factor $\left(\prod w\_{i} \right)^{\omega}$. For zero-dimensional systems, the complexity of Algorithm~\FGLM $nD^{\omega}$ (where $D$ is the number of solutions of the system) can be divided by the same factor $\left(\prod w\_{i} \right)^{\omega}$. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of $W$-degree $(d\_{1},\dots,d\_{n})$, these complexity estimates are polynomial in the weighted B\'ezout bound $\prod\_{i=1}^{n}d\_{i} / \prod\_{i=1}^{n}w\_{i}$. Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by the weighted Macaulay bound $\sum (d\_{i}-w\_{i}) + w\_{n}$, and this bound is sharp if we can order the weights so that $w\_{n}=1$. For overdetermined semi-regular systems, estimates from the homogeneous case can be adapted to the weighted case. We provide some experimental results based on systems arising from a cryptography problem and from polynomial inversion problems. They show that taking advantage of the weighted homogeneous structure yields substantial speed-ups, and allows us to solve systems which were otherwise out of reach

Factoring Polynomials and Groebner Bases

Factoring polynomials is a central problem in computational algebra and number theory and is a basic routine in most computer algebra systems (e.g. Maple, Mathematica, Magma, etc). It has been extensively studied in the last few decades by many mathematicians and computer scientists. The main approaches include Berlekamp\u27s method (1967) based on the kernel of Frobenius map, Niederreiter\u27s method (1993) via an ordinary differential equation, Zassenhaus\u27s modular approach (1969), Lenstra, Lenstra and Lovasz\u27s lattice reduction (1982), and Gao\u27s method via a partial differential equation (2003). These methods and their recent improvements due to van Hoeij (2002) and Lecerf et al (2006--2007) provide efficient algorithms that are widely used in practice today. This thesis studies two issues on polynomial factorization. One is to improve the efficiency of modular approach for factoring bivariate polynomials over finite fields. The usual modular approach first solves a modular linear equation (from Berlekamp\u27s equation or Niederreiter\u27s differential equation), then performs Hensel lifting of modular factors, and finally finds right combinations. An alternative method is presented in this thesis that performs Hensel lifting at the linear algebra stage instead of lifting modular factors. In this way, there is no need to find the right combinations of modular factors, and it finds instead the right linear space from which the irreducible factors can be computed via gcd. The main advantage of this method is that extra solutions can be eliminated at the early stage of computation, so improving on previous Hensel lifting methods. Another issue is about whether random numbers are essential in designing efficient algorithms for factoring polynomials. Although polynomials can be quickly factored by randomized polynomial time algorithms in practice, it is still an open problem whether there exists any deterministic polynomial time algorithm, even if generalized Riemann hypothesis (GRH) is assumed. The deterministic complexity of factoring polynomials is studied here from a different point of view that is more geometric and combinatorial in nature. Tools used include Gr\u27{o}bner basis structure theory and graphs, with connections to combinatorial designs. It is shown how to compute deterministically new Gr\u27{o}bner bases from given G\u27{o}bner bases when new polynomials are added, with running time polynomial in the degree of the original ideals. Also, a new upper bound is given on the number of ring extensions needed for finding proper factors, improving on previous results of Evdokimov (1994) and Ivanyos, Karpinski and Saxena (2008)

Closeout Report Department of Energy Grant DE-FG02 95ER40931 Advanced Map Methods for the Description of Particle Beam Dynamics

The above grant was active at Michigan State University from 1994 until 2007. We summarize and document the various activities and key output under the grant, including degrees awarded to graduate students at MSU and through the VUBeam program sponsored by the grant, the books, publications and reports produced, the meetings organized, and the presentations given