Effective Localization Using Double Ideal Quotient and Its Implementation

  title={Effective Localization Using Double Ideal Quotient and Its Implementation},
  author={Yuki Ishihara and Kazuhiro Yokoyama},
In this paper, we propose a new method for localization of polynomial ideal, which we call “Local Primary Algorithm”. For an ideal I and a prime ideal P, our method computes a P-primary component of I after checking if P is associated with I by using double ideal quotient (I : (I : P)) and its variants which give us a lot of information about localization of I. 

Modular techniques for effective localization and double ideal quotient

This paper applies modular techniques effectively to computation of such double ideal quotient and its variants, where first the authors compute them modulo several prime numbers and then lift them up over rational numbers by Chinese Remainder Theorem and rational reconstruction.

Computation of a Primary Component of an Ideal from Its Associated Prime by E ff ective Localization

This is an enhanced full paper version of [Ishihara-Yokoyama, 2018] and contains detailed proofs, additional examples and new algorithms. In [Ishihara-Yokoyama, 2018], we proposed e ff ective methods

Modular Techniques for Intermediate Primary Decomposition

An algorithm for ''Strong Intermediate Primary Decomposition" via maximal independent sets by using modular techniques, utilizing double ideal quotients to check whether a candidate from modular computations is an intersection of prime divisors or not is devised.

Constructive arithmetics in Ore localizations enjoying enough commutativity



Localization and Primary Decomposition of Polynomial Ideals

A new method for primary decomposition of a polynomial ideal, not necessarily zero-dimensional, is proposed and a detailed study for its practical implementation is reported on.

Direct methods for primary decomposition

SummaryLetI be an ideal in a polynomial ring over a perfect field. We given new methods for computing the equidimensional parts and radical ofI, for localizingI with respect to another ideal, and

Gröbner Bases and Primary Decomposition of Polynomial Ideals

Primary Decomposition: Algorithms and Comparisons

Three results are obtruned, which provide partial remedies for shortcomings in Hilbert series and degree bounds in the modular case and are a generalization of Goobel’s degree bound to the case of monomial representations.

A Singular Introduction to Commutative Algebra

From the reviews of the first edition: "It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in

Computational methods in commutative algebra and algebraic geometry

  • W. Vasconcelos
  • Mathematics
    Algorithms and computation in mathematics
  • 1998
The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation.

Introduction to commutative algebra

* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings *

Solving systems of polynomial equations

  • D. Manocha
  • Computer Science, Mathematics
    IEEE Computer Graphics and Applications
  • 1994
The author presents an algorithm for solving polynomial equations using the combination of multipolynomial resultants and matrix computations, which is efficient, robust and accurate.

Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics

  • Avalon Publishing,
  • 1994