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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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 ...
Tuong Nguyen Minh's user avatar
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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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$\tilde f_i = f_i + g_i$, then $\dim \mathbb C[t_1,\dots, t_d]/\langle f_i \rangle \leq \dim \mathbb C[t_1,\dots,t_d]/ \langle \tilde f_i \rangle$

Let $\mathbb C[t_1,\dots, t_d]$ be the multivariate polynomial ring with a certain monomial order $\leq$. I will denote monomials by $x^\alpha$, where $\alpha \in \mathbb N^d$. For $f = \sum_\alpha a_\...
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Uniqueness of quotients when reducing with Gröbner basis

Let $K$ be a field, and let $G = (g_1, \ldots, g_m)$ be a Gröbner basis in $K[x_1, \ldots, x_n]$ (i.e. $G$ is a Gröbner basis for the ideal it generates). By Adams, Loustaunau - An Introduction to ...
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Finding implicit form of Nyquist curve without using a general algorithm for Groebner bases.

According to Conversion Methods Between Parametric and Implicit Curves and Surfaces, by Christoph N1. Hoffmann p.3 Every plane parametric curve can be expressed as an implicit curve. In problems ...
Math Keeps Me Busy's user avatar
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Under what conditions is $\text{in}_{<}(I)$ a reduced Grobner basis?

Let $I \subseteq K[x_1,...,x_n]$ be a graded ideal. Under what conditions is $\text{in}_{<}(I)$ a reduced Grobner basis? Here $\text{in}_{<}(I)$ is the ideal: \begin{align*} \text{in}_{<}(I)...
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Groebner basis over rational vs finite field

Some algorithms for calculating a Groebner basis are optimized for calculating with coefficients in a finite field. Having determined the basis over a finite field, I'd like to understand what ...
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Polynomial constraints for restricting: $a=0$ if and only if $b\neq 0$

For this discussion, I will be considering polynomials over multiple complex variables, and a system of polynomial constraints, where the constraints on the variables can be written as a set of ...
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Algorithm to simplify division of two multivariate polynomials

I'm looking for al algorithm to try to implement simplification of a fraction of two multivariate polynomials. Here's an example of one: $$\frac{(3xy^2+x^2y^2+7yx^2+21xy+2y^2+12x^2+14y+36x+24) }{(4xy^...
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Subring of polynomial rings generated by a set

I am thinking the generated subring of polynomial ring $K[X]$ where $K$ is a field. First I find that it is isomorphism to a quotient ring of $Z[Y]$. For example, let $S = \{f_1,f_2\}$ and $f_1, f_2\...
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Proving that an ideal is primary

Problem 1: Show that the ideal $$I:=(af+d^2,bf+de,cf+e^2)+(a,b,c)^2+(a,b,c)(d,e)$$ is primary in the polynomial ring $k[a,b,c,d,e,f]$ where $k$ is a field. Macaulay2 confirmed the answer. But I ...
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$ n $ quadratic forms in $ n+1 $ variables

This answer seems to imply that there is something special about a system of $ n $ quadratic forms $ q_1, \dots, q_n $ in $ n+1 $ unknowns $ x_1, \dots, x_{n+1} $. I want to understand better why this ...
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If $G$ is a Gröbner basis of $I$ with respect to $<_w$, then $\{in_w(g)\ |\ g\in G\}$ is a Gröbner basis for $in_w(I)$ with respect to $<$

Let $I \subset R = K[x_1,...,x_n]$ be an ideal and $<$ a monomial term ordering on $R$. Let $w\in \mathbb{R}_{\geq 0}^n$ be a weight vector. Then for each $f = \sum c_\alpha x^\alpha \in I$ we can ...
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Declaring constants in Sagemath

I want to find the Groebner base of a ideal,the ideal is generated by some polynomials with constant coefficients, but they do not have numerical values. ...
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Strange Sage behavior grobner bases

I have the following code in Sage ...
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Applications of Gröbner bases for beginners

What are some applications of Gröbner bases that could be interesting to a group of students that more or less only studied Chapter 1 and Chapter 2 of Ideals, Varieties, and Algorithms by David A. Cox ...
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How to compute the intersection of ideals $(w,y,xz+x+z), (w,x,yz+y+z), (z,y,xw+x+w), (z,x,yw+y+w), (x+y,zw+z+w,y^2)$ over $\mathbb{F}_p$?

Let $p$ be any prime number in $\mathbb{Z}$, and consider the ideals $(w,y,xz+x+z)$, $(w,x,yz+y+z)$, $(z,y,xw+x+w)$, $(z,x,yw+y+w)$, $(x+y,zw+z+w,y^2)$ in $\mathbb{F}_p[x,y,z,w]$. My question is, how ...
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Does ideal reduction commute with intersection?

Let $\mathbb{Z}[T_1,...,T_n]$ be the ring of polynomials over $\mathbb{Z}$, and $\mathbb{F}_p[T_1,...,T_n]$ be the ring of polynomials over $\mathbb{F}_p$, with the canonical projection $p:\mathbb{Z}[...
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How to prove using Groebner Bases that $x^2 +y^2 = −1, x^3 +y^3 = −1, x^5 +y^5 = −1$ is inconsistent in $\Bbb C\;^2$?

How can it be proved using Groebner Bases that the following system of equations is inconsistent in $\Bbb C\;^2$ ? $x^2 +y^2 = −1, x^3 +y^3 = −1, x^5 +y^5 = −1$
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How is the reduced Groebner base of the ideal of an inconsistent polynomial system of equations?

Given a polynomial system ($f_1=0,....,f_s=0$) and the ideal $I$ <$f_1,....,f_s$>. How is the reduced Groebner base of $I$ in the case that the system has no solutions?
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Would this be correct to prove that two ideals are equal? [closed]

Given that reduced Gröbner bases are unique for any given ideal and any monomial ordering, would it be correct to prove that two given ideals, $I_1$ and $I_2$, are equal following this process? ...
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Polynomial Ideal with fixed basis

I'm looking for the algorithms associated with the two Magma functions PolynomialWithFixedBasis and Coordinates. In particular, if $g_1,\ldots,g_n$ is a Gröbner Basis (in decreasing order), I would ...
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Name of theorem / result related to Grobner Bases

What is the name of the following theorem / result ? There exist $x_1, x_2 ... x_n$ in $\mathbb{C}$ such that $f_i(x_1, x_2 ... x_n) = 0$ for each $i$ between $1$ and $k$ (i.e the variety is not null)...
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Finding the number of solutions of a system of multivariate polynomials without solving the system.

I have a system of multivariate polynomial equations, say for 3 variables, $$ f_1(x,y,z)=0 \ , \quad f_2(x,y,z)=0 \ , \quad f_3(x,y,z)=0 $$ I need to find the number to solutions to this problem. I am ...
Giulio Crisanti's user avatar
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A radical equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$

Solve the equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$. I am looking for real roots. The graph of the equation tell us there are 4 solutions: roughly at $\pm0.09, \pm 1.64$, but I want to ...
Sean Ian's user avatar
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How to solve this system of multivariate polynomial equations for $0<x_7<x_6<x_8 \le 1$? Groebner basis maybe?

I am reformulating my question according to the guidelines I was given. I have the following problem: I cannot find a way to solve the system of equations further down. This is the calculations from ...
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How to find roots of a system of multivariate polynomials? [closed]

I am trying to find the roots of a system of 3 multivariate polynomials with 3 variables. The polynomials are really 'ugly'. So far I have tried to find a Groebner Basis in Maple and got a Groebner ...
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Is a monomial basis a Groebner basis?

Given a monomial ideal $$I = \langle x^{\alpha} \mid \alpha \in A \subset \Bbb Z^n_{\geq 0}\rangle,$$ I want to know if the rest of the polynomial division of $f \in \mathbb{K}[x_1,...,x_n]$ by the ...
user996159's user avatar
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Gröbner Basis consisting of 3 polynomials with exactly 13 common solutions

I'm studying Gröbner bases and am using the following definitions and results: Let $I$ be an ideal in $R = K[x_1,...,x_n]$ for some field $K$ and some $n>0$ and fix a monomial ordering on $R$. A ...
noparadise's user avatar
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Groebner basis for system of integral quadratic forms

I have a system of quadratic forms (homogeneous polynomials of degree $ 2 $) with integer coefficients. Each quadratic form is the trace of a product of matrices. I'm solving for the matrix entries. ...
Ian Gershon Teixeira's user avatar
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Are zero dimensional Ideals radical?

I have a question about Theorem 3.7.25. of Computational commutative algebra I by M. Kreuzer and L. Robbiano. Let $K$ be a perfect field, $I \subseteq K[x_1, \ldots, x_n]$, be a zero dimensional ...
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Finding redundant equations in a underdetermined system of multivariate polynomial equations over $\Bbb R$

Starting from a geometric problem, I came up with a system of multivariate (many lines and points) polynomial equations where some equations are redundant (because they correspond to redundant ...
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Non uniqueness of solution to system of polynomial equations using Gröbner bases

Consider the system of polynomial equations \begin{align} x + y + z &= a,\\ xy &= b,\\ z&=c, \end{align} where $x, y, z$ are the unknowns, and $a,b,c$ are known real numbers. It is clear ...
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Property of remainders of Grobner basis

Let $G$ be a Gröbner basis for an ideal $I \subseteq \mathbb{F}[x_{1}, \ldots x_{n}]$. Let $f,g$ be in $\mathbb{F}[x_{1}, \ldots x_{n}]$ and recall that $\overline{f}^{G}$ is the remainder of $f$ ...
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Determining reduced Groebner basis of $I(V)$ by using Lagrange

I am working on the following task from Ideals, Varieties, and Algorithms by Cox, Little, Shea: I already completed parts a and b of the task. But I struggle finishing part c: Using the hint I take $...
MaxwellDgt's user avatar
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Algorithm for writing an image of a polynomial in a quotient ring in terms of a given basis of the quotient ring.

I need to calculate the equivariant Chern classes of certain vector bundles on the classifying spaces of complex algebraic groups. In order to do this I am looking for a way to do the following ...
Maksym Dolgikh's user avatar
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Solve system of polynomial equations based on distance?

I am trying to find the solution for this polynomial system. For some context it represents a game where every player has to choose a location in the x-line, given that it is best to be closer to 0, ...
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Example of initial ideal

I have a problem with understand some example, which I present below. Let ideal $I = (x_1^2 + 3x_1x_2, 2x_1^2 + x_2^2)$. The initial monomial of both generators is $x_1^2$. However, twice the first ...
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Find (a generating set for) $\mathbb{Q}[x]\cap I$ where $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ (generate gröbner basis).

Consider the polynomial ring $\mathbb{Q}[x,y]$ and the ideal $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ in $\mathbb{Q}[x,y]$. $G=(x^2-y,y^2-x)$ is a (reduced) gröbner basis for $I$ wrt. graded ...
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When does the system $x^n-1=y^n-1=x(y-1)^2+y(x-1)^2=0$ have only the trivial solution $(1,1)$ over the prime field of order $p$?

Let $F_p$ be the prime field of order $p$, $\overline{F_p}$ be its algebraic closure, and $n$ be an integer such that $\gcd(n,p)=1$. Consider the following three polynomials in $F_p[x,y]$: $$ p_1(x,y)=...
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Extension theorem over reals

Is there an equivalent of the following theorem from Cox, Little & O'Shea over reals? Definition. Given $I=\left\langle f_{1}, \ldots, f_{s}\right\rangle \subseteq k\left[x_{1}, \ldots, x_{n}\...
12345's user avatar
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How to find an ideal of variety?

I've been going through Ideals, Varieties, and Algorithms book by D. Cox et al., and have been stuck on exercise 4.12. I've solved it in a half, but have stuck on the next question: how to find an ...
Pavel Snopov's user avatar
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Is the induced map $V(I) \longrightarrow V(J)$ is surjective for $I=(xy-1,y^2-z)\subset\Bbb C[x,y,z]$ and $J=I\cap \Bbb C[y,z]$?

Let $\mathbb{C}[x, y, z]$ be a polynomial ring, and set $I = (xy - 1, y^2 - z)$. Compute $J = I \cap \mathbb{C}[y, z]$. Is the induced map $V(I) \longrightarrow V(J)$ surjective? I have trouble ...
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$G$ is a Grobner basis if and only if $\overline{f}^{G} = 0$

Claim: $G = \{g_1, \cdots, g_s\}$ is a Grobner basis if and only if for all $f \in I \subseteq k[x_1, \cdots, x_n]$ we have $\overline{f}^{G} = 0$. Here, the overline notation means remainder upon ...
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Number of monomials in the complement of $\langle LT(I) \rangle$.

I'm trying to find the number of monomials in the complement of $\langle LT(I) \rangle$, with both the lexicographic ($>_{lex}$) and graduated lexicographic ($>_{grlex}$) orders for the ideal $I ...
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