41,769 research outputs found
An algorithm for computing the integral closure
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
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
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
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
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
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
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
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