Questions tagged [groebner-basis]
A Gröbner basis is a type of a generating set of an ideal in a polynomial ring over a field. It is a multivariate non linear generalization of Gaussian elimination and Euclid's algorithm.
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Differences in meaning for notations $\alpha_i$ versus $\alpha(i)$ and meaning of $\beta(ij)$ for denoting axioms in monomials
The following are partly taken from Malik and Sen's Fundamentals of Abstract Algebra
Background
First we note that we can reconstruct the monomial $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ from $...
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How to break symmetry of a polynomial ideal to simplify Groebner basis?
I have an ideal $I$ generated by a set of polynomials $\{ p_i \}$. There are some variable permutations to which the ideal is symmetric. By this I mean (apologies if there is a standard term for this) ...
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How to express the originals system of equation in terms of its Groebner bases?
As a network engineer I need to explain some mathematical stuff to my fellow coleagues. Particularly, I need to explain the fact the the Groebner Basis will create an equivalent system. One particular ...
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Bounds on growth of Gröbner bases
Let $I$ be an ideal with $k$ generators i.e. $I = \langle f_1, \dots, f_k\rangle$ where $f_1, \dots, f_k \in k[x_1, \dots, x_n]$ and fix a monomial ordering.
I am interested in what happens to the ...
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How to compute $\operatorname{Length}(k[[x_1,\dots,x_n]]/I)$ for some ideal $I$
Let $k$ be an algebraically closed field and $\mathcal{O}_0=k[[x_1,\dots,x_n]]$ be the ring of formal power series, $I$ be an ideal of $\mathcal{O}_0$ such that $\operatorname{Spec}(\mathcal{O}_0/I)$ ...
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Example of Grauert's Division Theorem
We are trying to compute an explicit example of Grauert's Division Theorem, as it appears in De Jong and Pfister's Local Analytic Geometry book.
In particular we are trying to compute the following ...
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Why does the F4 Algorithm return a Gröbner Basis?
I am currently trying to understand the F4 algorithm. I am working with the book „Ideals, Varieties and Algorithms“ from Cox et al and have problems with understanding their proof for the correctness ...
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solving a system equations with Groebner basis
We have the equations:
$$ x+y+z=3\\x^2+y^2+z^2=5 \\x^3+y^3+z^3=7 $$
Using Groebner bases techniques, i want to find : $$x^5+y^5+z^5 \\and\\ x^6+y^6+z^6$$
\
What I did was to find a Groebner basis, in ...
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Method for solving polynomial system without multilinear form?
I am an engineer who is currently working with some network optimization problem during my post graduate study. During my study time, I see that sometimes I need to look for solution of polynomial ...
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Theorem 1.2.7 in Sturmfel's Algorithms in Invariant Theory: Hilbert Series and Monomial Ideals
This is a question on a proof of Theorem 1.2.7 in Sturmfel's Algorithms in Invariant Theory. I will restate the whole proof up to where I am confused.
First some notation.
Let $\sigma_i(x_1, \ldots, ...
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Determining the colon ideal in a polynomial ring
I am a grad student having background in a Commutative Algebra. I need help with the following.
Let
$$
\begin{aligned}
I =
\langle
&x_1^4,\, x_2^4,\, x_3^4,\, x_4^4, \\
&x_1^3 x_2^3,\, x_1^...
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Proving membership of monomials to an ideal [closed]
I am a grad student having background in Algebra. I need help with the following.
Let $I=\langle x_1^3,x_2^3,x_3^3,x_1 x_2 x_3,x_1^2 x_2^2, x_1^2 x_3^2, x_2^2 x_3^2 \rangle$ be an ideal in a ...
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Gröbner basis and dg structures
Gröbner basis is typically defined for ideals of polynomial rings over a field and there are several generalizations/extensions of this notion for non-commutative structures or differential algebras.
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compute Matrix relative to monomial Basis
I have the radical Ideal of $\mathbb{C}$ [x,y,z] generated by $\{x - 3 y - z + 9, z^2 -3z + 2, yz -2y - 3z + 6, y^2 - 5y + 6\}$ which form a Gröbner basis with TO deglex (x>y>z) and I want to ...
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Trying to understand basic Gröbner basis theorem
The following is from some study material I was provided. I will interject at the parts that have stumped me.
Theorem
Let < be a monomial order on the polynomial ring K[X].
Let ⟨0⟩ ≠ I ⊂ K[X] be an ...
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