41,769 research outputs found

    An algorithm for computing the integral closure

    Get PDF
    In this article we give an algorithm for computing the integral closure of a reduced Noetherian ring R, in case this integral closure is finitely generated over R.Comment: LaTeX2

    An algorithm for computing the integral closure

    Full text link
    We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field

    Integral D-Finite Functions

    Full text link
    We propose a differential analog of the notion of integral closure of algebraic function fields. We present an algorithm for computing the integral closure of the algebra defined by a linear differential operator. Our algorithm is a direct analog of van Hoeij's algorithm for computing integral bases of algebraic function fields

    An Algorithm for Computing the Ratliff-Rush Closure

    Full text link
    Let I\subset K[x,y] be a -primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I= whenever m_{i} is contained in the integral closure of the ideal . This generalizes of the work of Crispin \cite{Cri}. Also, it provides generalizations and answers for some questions given in \cite{HJLS}, and enables us to construct infinite families of Ratliff-Rush ideals

    Conormal Spaces and Whitney Stratifications

    Full text link
    We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to L\^e and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.Comment: There is an error in the published version of the article (Found Comput Math, 2022) which has been fixed in this update. Section 3 is entirely new, but the downstream results Sections 4-6 remain largely the same. We have also updated the Runtimes and Complexity estimates in Section 7. The def. of the integral closure of an ideal has also been correcte

    Algorithms for Galois extensions of global function fields

    Get PDF
    In this thesis we consider the computation of integral closures in cyclic Galois extensions of global function fields and the determination of Galois groups of polynomials over global function fields. The development of methods to efficiently compute integral closures and Galois groups are listed as two of the four most important tasks of number theory considered by Zassenhaus. We describe an algorithm each for computing integral closures specifically for Kummer, Artin--Schreier and Artin--Schreier--Witt extensions. These algorithms are more efficient than previous algorithms because they compute a global (pseudo) basis for such orders, in most cases without using a normal form computation. For Artin--Schreier--Witt extensions where the normal form computation may be necessary we attempt to minimise the number of pseudo generators which are input to the normal form. These integral closure algorithms for cyclic extensions can lead to constructing Goppa codes, which can correct a large proportion of errors, more efficiently. The general algorithm we describe to compute Galois groups is an extension of the algorithm of Fieker and Klueners to polynomials over function fields of characteristic p. This algorithm has no restrictions on the degrees of the polynomials it can compute Galois groups for. Previous algorithms have been restricted to polynomials of degree at most 23. Characteristic 2 presents additional challenges as we need to adjust our use of invariants because some invariants do not work in characteristic 2 as they do in other characteristics. We also describe how this algorithm can be used to compute Galois groups of reducible polynomials, including those over function fields of characteristic p. All of the algorithms described in this thesis have been implemented by the author in the Magma Computer Algebra System and perform effectively as is shown by a number of examples and a collection of timings

    Computing invariants of algebraic group actions in arbitrary characteristic

    Full text link
    Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]^G in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes K[X]^G in terms of a so-called colon-operation. From this, generators of K[X]^G can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is K[X]^G. Along the way, we develop some techniques for dealing with non-finitely generated algebras. In particular, we introduce the finite generation locus ideal.Comment: 43 page
    • …
    corecore