1,720,976 research outputs found
Integral closure of Noetherian rings
No full textAfter giving a proposition which reduces the problem of computing the integral closure of a general noetherian ring to the three problems: Compute a universal denominator d (element in the conductor). Compute radical of the ideal generated by d. Compute ideal quotients. We show that for the common case of affine domains, i.e. domains which are finitely generated over fields, of characteristic zero, we can use an effective localization in order to perform most of the computation in one dimensional rings where it can be done with linear algebra
Square-free Algorithms in Positive Characteristic
No full textWe study the problem of the computation of the square-free decomposition for polynomials over fields of positive characteristic. For fields which are explicitly finitely generated over perfect fields, we show how the classical algorithm for characteristic zero can be generalized using multiple derivations. For more general fields of positive characteristic one must make an additional constructive hypothesis in order for the problem to be decidable. We show that Seidenberg's Condition P gives a necessary and sufficient condition on the field K for computing a complete square free decomposition of polynomials with coefficients in any finite algebraic extension of K
Computation of the radical of polynomial ideals over fields of arbitrary characteristic
No full textWe study the problem of the computation of the radical of an ideal of polynomials with coefficients over fields of arbitrary characteristic. We show how to use Seidenberg's condition P to solve this problem in the case of positive characteristic
Generators of the ideal of an algebraic space curve
No full textIn this paper we show that the ideal of any algebraic curve in affine 3-space
whose Jacobian matrix has rank at least 1 at every singular point of the curve can be generated by three polynomials and we give
constructive procedures to compute such generators
Degree Reduction under Specialization
No full textWe examine the degree relationship between the elements of an ideal I ⊆R[x] and the elements of f(I ) where f: R → R is a ring homomorphism. When R is a multivariate polynomial ring over a field, we use this relationship to show that the image of a Greobner basis remains a Greobner basis if we specialize all the variables but one, with no requirement on the dimension of I . As a corollary we obtain the GCD for a collection of arametric univariate polynomials.
We also apply this result to solve parametric systems of polynomial equations and to reexamine the extension theorem for such systems
Derivations and radicals of polynomial ideals over fields of arbitrary characteristic
No full textThe purpose of this paper is to give a complete effective solution to the problem of computing radicals of polynomial ideals over general fields of arbitrary characteristic. We prove that Seidenberg’s “Condition P" is both a necessary and sufficient property of the coefficient field in order to be able to perform this computation. Since Condition P is an expensive additional requirement on the ground field, we use derivations and ideal quotients to recover as much of the radical as possible. If we have a basis for the vector space of derivations on our ground field, then the problem of computing radicals can be reduced to computing p th roots of elements in finite dimensional algebras
Irreducible decomposition of polynomial ideals
No full textIn this paper we present some algorithms for computing an irreducible decomposition of an ideal in a polynomial ring R=K[x1,...,xn] where K is an arbitrary effective field
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