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By a result of Latimer and MacDuffee, there are a finite number of equivalence classes of n × n matrices over F q [ T ] with minimum polynomial p (X) , where p is an n th degree polynomial, irreducible over F q [ T ]. In this paper, we develop an algorithm for finding a canonical representative of each matrix class, for p (X) = X 2 − Γ X − Δ ∈ F q [ T ] [ X ]. [ABSTRACT FROM AUTHOR]

The GVW algorithm provides a new framework for computing Gröbner bases efficiently. If the input system is not homogeneous, some J-pairs with larger signatures but lower degrees may be rejected by GVW's criteria, and instead, GVW has to compute some J-pairs with smaller signatures but higher degrees. Consequently, degrees of polynomials appearing during the computations may unnecessarily grow up higher, and hence, the total computations become more expensive. This phenomenon happens more frequently when the coefficient field is a finite field and the field polynomials are involved in the computations. In this paper, a variant of the GVW algorithm, called M-GVW, is proposed. The concept of mutant pairs is introduced to overcome the inconveniences brought by inhomogeneous inputs. In aspects of implementations, to obtain efficient implementations of GVW/M-GVW over boolean polynomial rings, we take advantages of the famous library M4RI. We propose a new rotating swap method of adapting efficient routines in M4RI to deal with the one-direction reductions in GVW/M-GVW. Our implementations are tested with many examples from Boolean polynomial rings, and the timings show M-GVW usually performs much better than the original GVW algorithm if mutant pairs are found. [ABSTRACT FROM AUTHOR]

Abstract: Extensible (polynomial) lattice point sets have the property that the number N of points in the node set of a quasi-Monte Carlo algorithm may be increased while retaining the existing points. Explicit constructions for extensible (polynomial) lattice point sets have been presented recently by Niederreiter and Pillichshammer. It is the aim of this paper to establish extensibility for a powerful generalization of polynomial lattice point sets, the so-called hyperplane nets. [Copyright &y& Elsevier]

Abstract: In this paper we consider two problems, finding minimal polynomials over and over of a finite-length sequence over , in a lattice and its sublattice respectively using lattice theory over polynomial rings. And we deduce a relationship between them by a lattice basis transformation matrix. As a byproduct, we present a new synthesis algorithm for the first problem just using operations in instead of as in Berlekamp–Massey algorithm. [Copyright &y& Elsevier]

Abstract: Let p be prime and a degree n unramified extension of the ring of p-adic integers . In this paper we give an overview of some very fast deterministic algorithms for common operations in modulo . Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in , for most basic operations on these fields, including Galois conjugation, Teichmüller lifting and computing minimal polynomials. [Copyright &y& Elsevier]

Abstract: In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration. Niederreiter obtained an existence result for polynomial lattice rules modulo arbitrary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algorithms for polynomial lattice rules modulo an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy. In this work we provide construction algorithms for polynomial lattice rules modulo arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algorithm. We show how the search space of the Korobov-type algorithm can be reduced without sacrificing the convergence rate, hence this algorithm is particularly fast. Our findings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy. [Copyright &y& Elsevier]

Abstract: Guruswami–Sudan algorithm for polynomial reconstruction problem plays an important role in the study of error-correcting codes. In this paper, we study new better parameter choices in Guruswami–Sudan algorithm for the polynomial reconstruction problem. As a consequence, our result gives a better upper bound for the number of solutions for the polynomial reconstruction problem comparing with the original algorithm. [Copyright &y& Elsevier]