Questions tagged [monomial-order]

A monomial order is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication.

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An infinity of monomial order on $K[x,y]$ [closed]

I must prove that there are an infinity of monomial orders on $K [x, y]$. But I don't know where to start the demonstration
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Monomial order in $K[x,y]$

I'm learning abstract algebra and came across the following statement: In the ring $K [x,y]$ there is exactly a monomial order $≤$ with property as $y^i <x$ for any $i ≥ 2$. I'm thinking of ...
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Bijection between monomials in $k[x_1,\ldots,x_n]$ and $\mathbb{Z}^n_{\geq0}$

I am reading Cox, Little, O'Shea - Ideals, Varieties and Algorithms. Beginning the section on monomial orderings, the authors claim there is an obvious bijection between monomials in $k[x_1,\ldots,x_n]...
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Is there any convenient way to index the monomials up to a given order by a number?

One canonical way of indexing a monomial of $n$ variables is to use a $n$-tuple of the powers, i.e., using $(\alpha_1, \ldots, \alpha_n)$ to index $x_1^{\alpha_1}\ldots x_n^{\alpha_n}$. I'm wondering ...
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Uncountable many monomial orders for monomials with 2 variables

Why do we have for monomials with two variables x1, x2 uncountable many monomial orders? I could prove that the order defined by $x_1^ax_2^b ≤ x_1^cx_2^d$ is a monomial order if and only if $a + b\...
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Complete Intersection ideal whose initial ideal is not even Cohen-Macaulay

I am trying to find an example, such a example is asked in exercise 3.3 of the book Monomial Ideals (Herzog & Hibi). I would like to find a graded ideal $I\subset k[x_1,\cdots,x_n]$ which is ...
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51 views

Existance of monomial ordering

Let $k$ be a field and let $f\in k[X_1,\ldots,X_n]$ be a polynomial. Write $$f = \sum_\alpha \underline{X}^{\underline{\alpha}}\qquad \underline{X}^{\underline{\alpha}} \text{ is a monomial in }X_1,\...
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Need help in understanding definition of monomials

Wikipedia states two definitions of monomials- https://en.m.wikipedia.org/wiki/Monomial I have understood the first one but I am facing some problem in understanding the second one which says that-" ...
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A question about extending polynomial span to monomial basis

I have a final next week and our instructor gave us some examples with solutions but I could not understand some operations. Inner product is $$(p,q)=\int_{-1}^{1} p(t)q(t)dt$$ $W = Span\{1,t,t^2\}$ ...
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133 views

What is the meaning of degree compatible ordering?

Suppose I am working in a polynomial ring in several variables, say $k[X]=k[x_1,\dots, x_n]$. An ordering $<$ on $k[X]$ is said to be degree compatible, if: $\deg(X^u)<\deg(X^v) \implies X^u <...
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Is reverse lexicographic order the same as graded reverse lexicographic order?

I want to make sure whether the two monomial orderings are actually the same thing. I am confused because the Cox book on Ideals, Varieties and Algorithms mentions only the graded reverse ...