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THE MOVING CURVE IDEAL AND THE REES
THE MOVING CURVE IDEAL AND THE REES

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Algebras over a field

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... Example 2.2 from [10] gives an example of a non-Noetherian locally Noetherian domain D whose localizations DP are all Noetherian valuation domains. Noetherian valuation domains are Cohen-Macaulay, so this is an example of a non-Noetherian ring which is locally Cohen-Macaulay where locally Cohen-Maca ...


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Math 1300 Section 3.2 Notes 1 Operations with Polynomials
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On derivatives of polynomials over finite fields through integration

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... 37. OPEN ENDED Find two binomials that have 6x as one of the terms in their product. 38. CHALLENGE Find (x + 1) 2, (x + 2) 2, and (x + 3) 2. Is there a pattern in the products of binomials? If so, use the pattern to find (x + 6) 2 and (x + n) 2. 39. REASONING Does the product of two binomials always ...
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Pade Approximations and the Transcendence of pi

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Problem set 3 - Math Berkeley

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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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