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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
53 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 $I$ ...
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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 ...
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1answer
32 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 ...
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1answer
72 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 ...
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59 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 ...
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48 views

Universal Basis

How can I prove that $G=\{X-Y^2,XY-X,X^2-X,Y^2-Y^3\}$ is an universal Groebner Basis for the ideal $I=\{X-Y^2,XY-X\}$ in $\mathbb{Q}[X, Y]$ ? any suggestion is good. Thanks.
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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 ...
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1answer
38 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? ...
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1answer
37 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 ...
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191 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 $<$ ...
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98 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
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1answer
70 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 ...
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1answer
93 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 $v$. ...
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1answer
156 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 ...
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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 ...
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Calculating Grobner Bases of Subideals

Let $I = (f_1,\dots,f_r) \subset \mathbb{Z}[x_1, \dots, x_n]$. Further, let $J = (f_1, \dots, f_m)$ and $K = (f_{m+1}, \dots, f_r)$. Suppose that we know a Grobner basis for $I = (g_1, \dots, g_s)$ ...
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Calculating Grobner Bases of Subideals

Suppose we have an ideal $I = (f_1,\dots,f_k) \subset \mathbb{Z}[x_1, \dots, x_n]$. Let $J = (f_1, \dots, f_s) \subset I$. Suppose we knew that $I$ and $J$'s Grobner bases, under the standard ...
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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 ...
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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 ...
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1answer
88 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 ...
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1answer
42 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 } ...
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92 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) = ...
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1answer
34 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 ...
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1answer
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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 ...
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1answer
80 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 ...
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1answer
42 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 ...
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1answer
62 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 ...
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1answer
80 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 ...
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1answer
287 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 ...
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1answer
196 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 $J \cap \mathbb{Q}[y]$. ...
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127 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 ...
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1answer
157 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 ...
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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 $$ \begin{pmatrix} 2 & 2 \\ 1 & -2 \\ \end{pmatrix} $$ $$ ...
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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 ...
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273 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 ...
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1answer
160 views

Is there a reason why the $S$-polynomial is defined in this way?

In my book the $S$-polynomial of two nonzero polynomials $f$ and $g$ is defined as $$S(f,g) = \displaystyle\frac{x^w}{LT(f)} \cdot f - \frac{x^w}{LT(g)} \cdot g$$ where $\displaystyle x^w$ is the ...
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1answer
98 views

on systems of bivariate polynomial equations (quartic)

I need to find an analytical solution to a system of bivariate polynomials. Specifically: \begin{eqnarray} a_0 + a_1 x + a_2 y + a_3 xy+a_4 x^2 + a_5 y^2 + a_6 xy^2 + a_7 x^2 y + a_8 x^2 y^2 &= ...
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1answer
151 views

Computable Criteria to check whether a given basis is a Gröbner Basis

In an upcoming exam we have to do Gröbnber-Basis computation with Buchberger's algorithm. A typical example looks like this: $$ \langle f_1,f_2 \rangle $$ Then I compute the S-Polynomial ...
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2answers
487 views

Reduced Gröbner Basis

If have computed this Gröbner Basis with Buchberger's algorithm for Degree-Lexicographic-Ordering: $$\{ x^²y+x+1,xy^2+y+1,x-y \} $$ I want to to transform it into a unique representation form called ...
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1answer
69 views

Weighted initial ideal versus lex or graded reverse lex initial ideal

By imposing certain weights $\mathbf{w}$ on the variables, say, of a polynomial ring $k[x_1,\ldots, x_n]$, I read that we may obtain the initial ideal $in_{\mathbf{w}}(I)$ of an ideal $I$ with respect ...