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.

learn more… | top users | synonyms

2
votes
0answers
31 views

Translate a geometric theorem into polynomial equations — Theorem of the orthocenter of a triangle

This is Exercise 13 of Chapter 6 of Ideals, Varieties, and Algorithms by Cox et al. The problem asks to translate the following geometric theorem into polynomials and using Groebner basis to test ...
1
vote
0answers
28 views

What is a good lex order to compute the Groebner basis of this ideal?

This comes from chapter 6 of Ideals, Varieties and Algorithms by Cox et al. The equations are from a planar robot with three joints and one prismatic joint. See the following picture: Given a ...
0
votes
0answers
39 views

Grobner bases of a determinantal ideal

I've been studying algebraic geometry recently and there is a problem I'm struggling with: Suppose $A$ is a $m\times n$ complex matrix of rank $\leq r$, this is equivalent to all its $(r+1)\times ...
7
votes
0answers
75 views

What is the “projective limit” of a polynomial?

Bayer and Mumford, What can be computed in algebraic geometry, reads (in part): Let $S = k[x_0, \ldots, x_n]$ be the homogeneous coordinate ring of $\mathbb{P}^n$. [. . .] Choose a ...
3
votes
1answer
32 views

Show that the number of points of $V(I)$ is at most $m_1m_2…m_n$ if $x_i^{m_i}\in \left\langle \text{LT}(I) \right\rangle$.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Let $I\subset \mathbb{C}[x_1,...,x_n]$ be an ideal such that for each $i$, some power $x_i^{m_i}\in \left\langle ...
1
vote
1answer
59 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
1
vote
1answer
49 views

How can one show that an ideal with some property is zero-dimensional?

Let $\mathfrak{a}$ be an ideal in $\mathbb{k}[x_1, \ldots, x_n]$ and a Gröbner basis of the ideal be $\{g_1, \ldots, g_t\}$. For each $i = 1, \ldots,n$, there exists $j \in \{1, \ldots, t\}$ such that ...
1
vote
1answer
59 views

Help with computation and Gröbner basis

Hi guys I am learning a new software and a new topic (Gröbner basis) I have this problem $$ \begin{cases} 6-21(x_1x_2+x_1x_3+x_1x_4)=0 \\ 10-21(x_2x_1+x_2x_3+x_2x_4)=0 \\ ...
2
votes
1answer
35 views

Show that $G$ is a Groebner bases of $I$ if division of $f$ on $G$ is zero for all $f\in I$.

Let $I=\langle g_1,\dots, g_t\rangle$ be an ideal in $k[x_1,\dots,x_n]$ with $k$ a field. Let $G=\{g_1,\dots,g_t\}$. Show that if the remainder of $f$ on division by $G$ is $0$ for all $f\in I$, ...
0
votes
0answers
13 views

Prove Theorem with Groebner Basis

I'm trying to prove some theorems using Groebner Basis (as described in Cox, Little and O'Shea Link ) The mentioned book gives as an excercise to prove Pappus theorem using the given methodology, ...
2
votes
1answer
180 views

Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
1
vote
0answers
75 views

How to check if a polynomial is inside an ideal using a Groebner basis

I'm given that an ideal $I=\langle F_1, F_2, F_3, F_4, F_5, F_6, F_7\rangle$ $F_1=a+b+c-d-e-f$ $F_2=a+b+c-g-h-i$ $F_3=a+b+c-g-e-c$ $F_4=a+b+c-a-e-i$ $F_5=a+d+g-a-e-i$ $F_6=a+d+g-c-f-i$ ...
1
vote
1answer
99 views

Is the mentioned basis a Gröbner basis?

It's mentioned into my notes that if the ideal given as $I=\langle x+y+z, 3x-2y\rangle$, then $\{x+y+z, 5y+3z\}$ is a Gröbner basis for the ideal. I can see how $I=\langle x+y+z, 3x-2y\rangle=\langle ...
1
vote
0answers
48 views

Gröbner Basis and linear basis

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give ...
1
vote
0answers
71 views

How do I naively compute Gröbner bases?

I have a upcoming test tomorrow. In my class, my professor didn't give us complete proofs but taught us how to compute Gröbner base and he told me computing problems are gonna be on exam. I hate this ...
0
votes
0answers
38 views

Question about multiple solutions to a polynomial

Assume that $f(X,Y,Z,V,W)\in \mathbb{Z}[X,Y,Z,V,W]$ is some polynomial and assume that $f(x,y,z,v,w)=0$. I would like to know if there is some way to figure out if there are non-trivial constants in ...
0
votes
0answers
38 views

How to efficiently check whether two cubics are equivalent

I have a very long list of cubic polynomials in $N$ variables, with $N$ ranging from $2$ to $19$. For my purposes, any two cubics which are related by a rational change of basis in the $N$ variables ...
0
votes
0answers
48 views

A question about the size of reduced Groebner basis

Let $I=(f,g,h)$ be an ideal in the polynomial ring $k[x,y,z]$ with $LT(f)>LT(g)>LT(h)$ in the lexorder, and $I$ is "reduced" in the sense that $LT(g)\nmid LT(f),LT(h)\nmid LT(g),LT(h)\nmid ...
0
votes
1answer
70 views

Groebner basis of a maximal ideal

is it a true for a maximal ideal $I=\langle x-a,\,y-b\rangle$ the vector space $\mathbb{C}[x,y]/I$ always has the dimension one? I thought we would have a Groebner basis $G$ of the same form as $I$ ...
2
votes
0answers
35 views

Free resolution by Groebner basis

I am studying approaches of Groebner basis in Homological and commutative algebra. I am so confused how can I find the minimal resolution for the below ideal $$I=\langle ...
1
vote
1answer
47 views

What is the Implicitization Problem

Let $V$ be a subset of $k^n$ given parametrically as $x_1 =g_1(t_1,...,t_m) ...x_n=g_n(t_1,...,t_m)$. If the $g_i$ are polynomials (or rational functions) in the variables $t_j$, then $V$ will be an ...
2
votes
1answer
79 views

Let I, J ideals. Are they equal?

Let $$I= \langle 11x^5y+7xy^6+9,8xy^4+6xy+9 \rangle$$ $$J= \langle 7x^5y^2+17x^2y^5+29,13xy^4+62xy^3+19 \rangle$$ ideals. Examine whether those two ideals are equal. By seeing their 3D plots I ...
0
votes
1answer
65 views

Ideal equals the whole ring

Show that in the polynomial ring $S=K[x_1, ..., x_n]$, having an ideal $I = (g_1, ..., g_m)$ and ${g_1, ..., g_m}$ a Groebner bases of $I$, then $I = S$ if and only if one of the $g_i$ is a nonzero ...
2
votes
0answers
47 views

Parametric ideal decomposition

Let $x = \{x_{1},\dots, x_{n}\}$ be a set of variables and let $a = \{ a_{1}, \dots, a_{m}\}$ be a set of parameters. Let $\{f_{1}(a,x), \dots, f_{s}(a,x)\} \subset \mathbb{C}[a,x]$ be a set of ...
2
votes
1answer
45 views

Skew-Symmetric after base change symmetric?

Are there invertible matrices $A,B \in \textrm{GL}(\mathbb{C}^3)$ such that for every skew-symmetric matrix $S \in \textrm{Mat}_{3 \times 3} (\mathbb{C})$ the matrix $A \cdot S \cdot B$ is symmetric? ...
0
votes
1answer
44 views

find a basis and the dimensions of the solution space w

$$x+2y-2z+2s-t=0$$ $$x+2y-z+3s-2t=0$$ $$2x+4y-7z+s+t=0$$ I need to find the basis and dimensions. I'm not sure how to do it. The book I have doesn't have a very good example. I end up ...
0
votes
1answer
206 views

Intuition? how the author reaches the answer?

I've a question on 2 problems in this book: 2.4. Let $S = K[x_1, . . . , x_6]$. Let $f = x_1x_5 − x_2x_4$, $g = x_1x_6 − x_3x_4$ and $h = x_2x_6 − x_3x_5$. (a) Find a monomial order $<$ ...
3
votes
1answer
130 views

A monomial ideal, $I =\langle xy, xz, yz\rangle$, is radical

I need help in showing that $I =\langle xy, xz, yz\rangle$ is a radical ideal. Thanks
1
vote
1answer
90 views

Find a basis of E as a vector space over $ \mathbb{Q} $

Find a basis for the factor ring $$\frac{\mathbb{Q}}{<16x^4-30x^3+15x^2+6>} $$ as a vector space over $\mathbb{Q} $. I honestly don't even know how to start this :( I though I would use ...
2
votes
1answer
125 views

Calculating Grobner Bases

In this question, $ℚ[x,y,z]$ is endowed with the lexicographic order with $x > y > z$. Set $u:= x^2 + 2yz^2$ and $v:= y^2 - 3xz$. Denote by $J$ the ideal of $ℚ[x,y,z]$ generated by $u$ and $v$. ...
1
vote
1answer
194 views

How many ways are there to represent a monomial order, defined by $>$, by term order via matrices?

During the lecture, my professor brought up the list of project ideas to work on. One of the ideas I am interested and currently working on is term order via matrices. That is: I need to find the ...
4
votes
0answers
226 views

Computing toric ideals via saturation

I have recently got interested in toric varieties and I have a question concerning their ideals. Let $A \in \mathbb{Z}^{m \times n}$ and $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0 \}$. For any $u ...
3
votes
0answers
84 views

Buchberger's criterion to show Grobner basis for linear forms

Let $k$ be a field. A polynomial of the form $l=a_1x_1+\cdots+a_nx_n$ is called a linear form ($a_i\in k$), and its support is the set of all variables $x_i$ such that $a_i\neq 0$. Let $L\subseteq ...
1
vote
0answers
59 views

A confusion regarding Grobner bases.

A Grobner basis $\{g_1,g_2,\dots,g_r\}$ for an ideal $I$ is the set of polynomials such that $I=\langle g_1,g_2,\dots,g_r\rangle$. Also, if you take any polynomial in the ideal $I$, the leading term ...
2
votes
1answer
118 views

Prime ideals and sagbi bases

I'm trying to understand the following passage in Combinatorial Commutative Algebra by Miller and Sturmfels: If $R$ is any subalgebra of a polynomial ring that possesses a finite sagbi basis, then ...
2
votes
1answer
44 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
3
votes
1answer
96 views

Help with Gröbner bases

I have $f = XY+Y$ and $g = X^2 +1$ in $\mathbb{Q}[X,Y]$. Let $I =\langle f,g\rangle$ be the ideal in $\mathbb{Q}[X,Y]$ generated by $f$ and $g$. I have that $in_{<}(f) = XY$ and $in_{<}(g) = ...
1
vote
1answer
35 views

Show that for $f \in \mathbb Q[W,X]$ and $Q=f^G$ (the unique remainder), we have $f(W,X)=Q(WX,X^3)$ if $Q \in \mathbb Q[Y,Z]$.

Let $I = \langle WX-Y, X^3-Z \rangle \subset \mathbb Q[W,X,Y,Z]$, and $\le$ denote the lexicographic term ordering on $\mathbb N^4$ such that $W > X > Y > Z$. I've shown: i) The ...
0
votes
1answer
44 views

Use / take advantage of Gröbner basis $ G = (g_j)$ to write $f \in I = \langle f_i \rangle$ as a $k$-linear combination of the polynomials in $I$?

Let $R=k[X_1,...,X_n]$ be a polynomial ring, where $k$ is a field. Suppose we have a Gröbner basis $G = (g_1, g_2, ... , g_n), g_i \in R$ for the ideal $I = \langle f_1, f_2, ..., f_m \rangle, f_i ...
2
votes
1answer
82 views

Gröbner bases: Polynomial equations. Solution $x$ to $G \cap k[x_1, .., x_i]$ imply solution to $G \cap k[x_1, .., x_i, x_{i+1}]$, $x$ plugged in.

I'm have been studying Gröbner bases for a while now and seen a few examples in my textbook / exercises. Let $\mathcal k$ be a field and $\mathcal k[x_1,..,x_n]$ a polynomial ring. I wish to solve a ...
2
votes
1answer
45 views

If an ideal is made up by polynomials with disjoint variable parts, then those polynomials form a Grobner Basis.

I've been learning symbolic computation over the summer (just independent learning) and I'm at the section of my book about Grobner bases. There's an exercise I'd like to see a proof of, but have not ...
1
vote
1answer
85 views

degree of remainder on division of multivariate polynomials

Let $f, g_1, \cdots, g_s \in \mathbb{R}[x_1,\cdots,x_n]$ and consider the division of $f$ by the $g_i$. Standard multivariate division algorithm will give $f = \sum_i a_i g_i + r$. I have been trying ...
4
votes
1answer
86 views

Explicit generators of syzygies

Consider an $1\times n$ matrix $$ \mathbf{A}=\begin{pmatrix} f_1 &f_2 & \dots & f_n \end{pmatrix} $$ over $R=\mathbb{C}[X_1,\dots,X_r]$. Let $M=\oplus_{i=1}^n R\mathbf{e}_i$ be the ...
1
vote
1answer
449 views

How to check that given polynomials form a Groebner basis

I am wondering if some polynomials are given, how do we know whether they form Groebner basis or not. Note that it is not necessary that given poly's form a reduced Groebner basis. I know how to find ...
7
votes
1answer
212 views

Gröbner basis and generating set

I have come across the following past exam question... Define an ideal $J:=(z^2x+y^2-2y,x^3+y^3+z^3,x^2+2z^2) \subseteq \mathbb{Q}[x,y,z]. $ Compute a generating set for $J \cap \mathbb{Q}[y]$. ...
3
votes
1answer
139 views

Gröbner Basis for Ideal $J$

I have the following question... Consider the ideal $J:= (x^2y-x^2y^2,\ x^2z-z^2yx,\ x^2+xz) \subset \mathbb{Q}[x,y,z]$ Is $x \in J?$ Is $x \in \sqrt{J} $? I know finding if $x$ is in the radical ...
2
votes
1answer
178 views

Reduced Gröbner Basis for $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$

I have $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$ with lex $X>Y>Z $. I have calculated the Gröbner Basis as $G=\{ X^2+2XYZ, XY+2Y^2Z-1, X, 2Y^2Z-1 \}$. But the question I have asks for the Reduced ...
0
votes
2answers
127 views

Basis of matrices with a variable

So I have these bunch of matrices I want to find the value of a to find the basis $$ \begin{pmatrix} 2 & 2 \\ 1 & -2 \\ \end{pmatrix} $$ $$ ...
6
votes
1answer
49 views

Is there an ideal decomposition that counts the number of monomial generators?

Consider the ideal $I\subseteq S[x,y,z]$ where $S$ is some field of characteristic 0 (probably any field will do) and $I=\langle x^9-y^4z^4,y^9-x^5z^4,z^8-x^4y^5,x^6\rangle$. Notice that because the ...
3
votes
1answer
338 views

Dimension of a certain quotient ring of $\mathbb{C}[x_0,\ldots,x_{m-1}]$.

Let $A=\mathbb C[x_0,\dots,x_{m-1}]$ be the polynomial ring on $m$ variables. Define $X(u)=\sum_{i=0}^{m-1} x_i u^{i+1}$ and denote by $(X(u)^r)_n$ the coefficient of $u^n$ in the expansion of the ...