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5.1 Notes: Polynomial Functions monomial: a real number, variable
5.1 Notes: Polynomial Functions monomial: a real number, variable

Monomial regular sequences - Oklahoma State University
Monomial regular sequences - Oklahoma State University

... Nd containing Rd , there is a lex monomial space Ld such that |Ld + Pd | = |Nd | and |mLd + Pd+1 | ≤ |mNd |. Reorder the variables so that xi divides fi for all i. By Gröbner basis theory, we may assume (after taking an initial ideal if necessary) that Nd is a monomial space. Set N (0) = Nd . For e ...
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... completes the proof: use Chinese Remainder Theorem for this collection of ideals. Note that the least common multiple of (x − ai ) is their product, which is a polynomial of degree n. (7) Describe the quotient ring (i.e. find a simpler-looking ring isomorphic to it). Is the ideal (x2 + 1) maximal in ...
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Section 2.4: Real Zeros of Polynomial Functions

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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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