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Notes on Ring Theory
Notes on Ring Theory

Document
Document

Polynomial Factoring Algorithms and their Computational Complexity
Polynomial Factoring Algorithms and their Computational Complexity

The discriminant
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... f(x) is the same as solving the related polynomial equation, f(x) = 0. • Zero, solution, root ...
Lecture 1: Lattice ideals and lattice basis ideals
Lecture 1: Lattice ideals and lattice basis ideals

... We now give another interpretation of toric ideals. A subgroup L of Zn is called a lattice. Recall from basic algebra that L is a free abelian group of rank m ≤ n. The binomial ideal IL ⊂ S generated by the binomials fb with b ∈ L is called the lattice ideal of L. Consider for example, the lattice L ...
Factorization in Integral Domains II
Factorization in Integral Domains II

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POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS 1

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EFFECTIVE RESULTS FOR DISCRIMINANT EQUATIONS OVER

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The Power of Depth 2 Circuits over Algebras

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LEFT VALUATION RINGS AND SIMPLE RADICAL RINGS(i)

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3. Modules

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x - HCC Learning Web

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Factoring in Skew-Polynomial Rings over Finite Fields

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AES S-Boxes in depth

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Exercises. VII A- Let A be a ring and L a locally free A

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GENERALIZED CAYLEY`S Ω-PROCESS 1. Introduction We assume

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Honors Algebra 4, MATH 371 Winter 2010

< 1 ... 5 6 7 8 9 10 11 12 13 ... 30 >

Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field K[x1, ..,xn]. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, andGaussian elimination for linear systems.Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger's algorithm), by Bruno Buchberger in his Ph.D. thesis. He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work.However, the Russian mathematician N. M. Gjunter had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for local rings was developed independently by Heisuke Hironaka in 1964, who named them standard bases.The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras.
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