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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38 views

Multiplicative basis gets inherited for an algebra modulo a compatible ideal

Let $A$ be an associative $k$-algebra with unit. Definition. A set $B \subseteq A$ is called a multiplicative basis of $A$ if $B$ is a $k$-basis of $A$ and if $B \cup \{0\}$ is closed under ...
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Groebner basis property under evaluation of a variable [closed]

Be $G$ a Groebner basis for a monomial order O in the ring $F[x_{1},x_{2},\ldots x_{n}]$, with $F$ a field. And be $G_{x_{i}=k}$ the set of polynomials obtained from $G$ substituting the variable $x_{...
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Are minimal Groebner bases minimized bases?

With "minimal Groebner basis" I mean, fixed an ordering, a Groebner basis $G$ such that any proper subset of $G$ is no more a Groebner basis for the ideal $I(G)$ generated by $G$. With "...
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Doubt in computation of groebner basis and proving a set is a groebner basis.

I have a set $G$ , and I am trying to prove that the set is a groebner basis . From chapter $2$ of Ideals,Varieties and Algorithm I have the following theorem, A basis $G = \{g_1,\cdots, g_n \}$ for ...
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Buchberger algorithm with three variables

Consider $>_{degrevlex}$ with $x>y>z$ in $\mathbb{C}[x,y,z]$. Let $f_{1} = y^2 - xz$, $f_{2} = x^2y - z^2$, $f_{3} = x^3 - yz$. Show (manually) that $\lbrace f_{1}, f_{2}, f_{3}\rbrace$ is a ...
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Insights into the property of a “generic” matrix.

This is in fact related to the previous question that I have asked but I decided that it would be better if I post this as a new question 1)What is a generic matrix(the definitions in net dont really ...
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Confusion in grobner basis and ideals generated by matrices.

https://drive.google.com/file/d/1MzElOy1tgJGzZkCpzY8d_msKHuV-03kx/view?usp=sharing Here is a link to Strumfels 1990 paper(Grobner Basis and stanley decomposition of determinantal rings (pdf pg-137)). ...
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Doubt in the proof of “The Groebner Basis of the ideal $I_k$ with respect to the monomial ordering is the set $S_k$”

I am not sure how is the proof done. Notations used: $S_k$ is the set of of all $k \times k $ minors of the matrix $X$ with minors being of the form $[a_1 a_2 \cdots a_k \mid 1 \cdots k]$ (generic ...
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Help with computation in Gröbner Basis.

The problem is to compute the Gröbner basis of the following ideal using Buchberger's algorithm. Let $u=a+bx+cx^2+dx^3$, $v= a+by+cy^2+dy^3$, $w=a+bz+cz+dz^3 $. The ideal is $I= \langle a+bx+cx^2+dx^3,...
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Existence of reduced Groebner basis [closed]

In my class notes there is a Theorem saying: Every non zero ideal of the ring $F[x_1,\dots,x_n]$ has a reduced Groebner Basis. Unfortunately there is no proof. Can someone give the proof or refer to ...
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How is the replacement by the remainder gives a reduced Grobner basis?

Here is the proof I am stuck in understanding: But how is the replacement by the remainder gives a reduced Grobner basis?
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numerically solving a system of multivariate polynomials

I have a system of polynomial equations which I need to solve numerically. The one Im currently interested in has around 20 variables and a similar number of equations. All coefficients are rational ...
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Showing that $\{g_1, \dots , g_s\}$ is also a Gröbner basis for $M.$

Let $F$ be a free module (of finite rank) over $S = k[x_1, \dots , x_r]$ with monomial order >. Let $M \subset F$ be a submodule and let $B = \{g_1, \dots , g_t\}$ be a Gröbner basis for $M.$ I ...
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264 views

Reduced Gröbner basis for a submodule, existence and uniqueness.

Let $F$ be a free module (of finite rank) over $S = k[x_1, \dots , x_r]$ with monomial order >. Let $M \subset F$ be a submodule and let $B = \{g_1, \dots , g_t\}$ be a Gröbner basis for $M.$ I ...
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149 views

Reduced Gröbner bases are minimal.

Let $F$ be a free module (of finite rank) over $S = k[x_1, \dots , x_r]$ with monomial order >. Let $M \subset F$ be a submodule and let $B = \{g_1, \dots , g_t\}$ be a Gröbner basis for $M.$ I ...
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56 views

Normal form and remainder - Groebner Basis

I'm quite confused about the concept of normal form of a polynomial $f$ relative to the ideal $I$. It should be the remainder of the division of $f$ with a Groebner Basis of $I$. But does that mean ...
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Example of strict inclusion with respect to the product of initial ideals

I solved the following problem: Show that for any monomial order $>$ it is true that $\mathrm{in}_{>}(I)\mathrm{in}_{>}(J)\subseteq \mathrm{in}_{>}(IJ)$ for any two ideals $I,J$ of $k[x_{1}...
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Does this square polynomial system have a finite number of zeros?

Wish to know if this square polynomial system has a finite number of zeros: $$ \begin{align} f_1(z,w)&=-\frac{976 w^2}{53}+\frac{178 w z}{71}+\frac{27 w}{94}+\frac{323 z^2}{84}+\frac{271 z}{67}+\...
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Groebner Basis in Macaulay2

This is a follow-up question to this question (calling on @Jan-Magnus Okland for help). Plugging the following into Macaulay2: <...
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Using Groebner Basis to eliminte variables but end with empty set.

I want to infer the relationship between x and y (which is x == y) as the code (mathematica) ...
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Invert “matrix of polynomials” for a change of presentation of an Ideal

If I start with a polynomial ideal generated by a set of polynomials $f_i$, which is not a standard basis for the ideal, and then I obtain a standard basis for the ideal to be a set of polynomials $...
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How to remove roots from an equation?

The question here is how (if it is even possible) to remove the square root terms and transform the following equation to a polynomial with one unknown $x$. The coefficients $a$, $b$, $c$, and $d$ are ...
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$I=\langle x^2+2y^2-3, x^2+xy+y^2-3\rangle$ in $\Bbb Q[x,y]$ how do I find the Grobner basis for $I \cap \Bbb Q[x]$ and $I\cap\Bbb Q[y]$?

Given the ideal $$I=\langle x^2+2y^2-3, x^2+xy+y^2-3\rangle$$ in $ ℚ[x,y]$, how do I find the Grobner basis for $I \cap \Bbb Q[x]$ and $I\cap\Bbb Q[y]$? I have learned to find a Grobner basis using ...
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The best methods for multivariate polynomial equations over finite fields

I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given ...
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If $I=\langle q\rangle$ then $\{q\}$ is a Gröbner basis for I.

I've been reading Cox "Ideals, Varieties and Algorithms" and I've got stuck on this problem. It looks easy but I don't know what to do. I tried finding a contradiction on remainder being not ...
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Is a Gröbner basis after substitution still a Gröbner basis?

Suppose we have an ideal $I = (g_1,\dots,g_k)\subseteq k[x_1,\dots,x_n]$ where $g_1,\dots,g_k$ is a Gröbner basis. Let $(c_1,\dots,c_n)\in V(I)$ be a point and consider the ideal $I_n = (\overline{g_1}...
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Do Groebner bases over rationals lose information about complex solutions?

Given a system of polynomial equations, if I calculate the Groebner basis over the rationals and get {1} then: My understanding is that I can then conclude there are no rational solutions. Is this ...
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Reduction of a system of multivariate nonlinear polynomials

I am trying to find a way to simplify a set of multivariate polynomial equations. As an example, given the free variables $\mu_1,\mu_\star,x,\psi_1,\psi_2$, I am specifically interested in the ...
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Computing Groebner Basis with given generators and relations of a module

Suppose I have a module $M$ over a polynomial ring $R=k[X_1,\ldots,X_n]$, viewed as a submodule of $R^m$. I know the generators of $M$ $$ v_1=[a_{11},a_{12},\ldots,a_{1m}],\\ v_2=[a_{21},a_{22},\ldots,...
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Division Algorithm for non-global ordering in K[x,y]

I'm trying to adapt the devision algorithm to non-global weight-ordering. Given $f=x^2$, $G=\{x-1\}$ and the weight vector $v=(-1,1)$. Now $K[x,y]$ is adopted with a weight-ordering ($deg_v(x^\alpha*y^...
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How do you account for the nonnegativity of a polynomial when using Groebner basis methods?

One can account for a polynomial $f(x_1,\dots,x_n)$ being nonzero in a Groebner basis computation by adding the polynomial $tf(x_1,\dots, x_n)-1$ to the list of generators and eliminating the $t$. Is ...
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Given a multivariate polynomial ideal $A$ and another $B \subset A$, how do you compute an ideal $C$ such that $A=B+C$ and $B\cap C = \emptyset$?

This is a little analogous to finding the orthogonal complement of a linear subspace.
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Trouble understanding conclusion regarding Gröbner bases

I'm having trouble understanding how the underlined conclusion is made. In my head, it means that $$aX^v=d_1X^{v_1}(c_1+\frac{c_2d_2}{d_1}+\dots+\frac{c_rd_r}{d_1})X^{u_1},$$ but then $(c_1+\frac{...
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Gröbner basis of ideals

I have the following problem: Let $I, J ⊂ S $ be ideals and $<$ a monomial order on $S$. Let $G, G'$ be Gröbner bases of $I$, respectively $J$, with respect to $<$ . Prove that if $\,\...
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Help with a Groebner basis

I need to calculate a Grobner basis of $$I=(u-x^2,v-y^2,w-xy)$$ with lexicographic order $x>y>u>v>w$. I am trying to proceed with Buchberger's algorithm, but i probably didn't quite ...
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$f_m:=x^m+x^{-m^2},m\in\mathbb{Z}$ is not a $\mathbb{Q}$ basis for $\mathbb{Q}[x,x^{-1}]$

I want to show that $x$ cannot be expressed as a sum $\sum_{k\in\mathbb{Z}}q_kf_k,q_k\in\mathbb{Q}$ but I don't know how I can do this.
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Showing that the set $f_{\mu}$ with $\deg(f_\mu)= \mu$ is a $F$ basis for $F[x_1,…,x_n]$, $\mu\in\mathbb{N_0^n}$

For every $\mu\in\mathbb{N_0^n}$ let $f\in F[x_1,...,x_n]$ where F is a fiedl such that $\deg(f_\mu)=\mu$ where the degree is defined for an accepted order. Show that $f_\mu,\mu\in \mathbb{N_0^n}$ is ...
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Biggest circle under a polynomial curve

What is the biggest circle that is contained in the region bounded by the graph of the polynomial $f(x) = x(1-x)(2x+1)$ and the x-axis interval $[0, 1]$? (Here's the thing in Desmos) Here's what I ...
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Is this polynomial equation redundant?

Consider $m$ real, symmetric, positive definite matrices $P_1,\dots,P_m$ of dimension $n\times n$. Now let $x\in\mathbb{C}^n$ with $m<n$ and consider the $m$ equations $x^TP_ix=1, \ i=1\dots,m$ to ...
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Unexpected result for Groebner basis

I am trying to (re)compute the Groebner basis used for proving the theorem of Pappus (plane geometry) as done here on pages 22-23. Since I have no Mathematica license, I used e.g. REDUCE: ...
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Is a minimal Gröbner Basis a minimal system of generators?

I'm studying Gröbner Bases and I'm trying to show that every finitely generated graded module $M$ over $k[x_1,\ldots,x_n]$ has a minimal graded free resolution of length $l \leq n$. (According to ...
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Let $P$ be prime and $Q\subset P$. How feasible is the computation of a Gröbner basis of a min prime over $Q$ compared to the computation for $P$?

Context: I am computing the reduced Gröbner bases with respect to degRevLex for the following two ideals: $P$ is a homogeneous prime ideal with $36$ generators $\{g_1,\dots,g_{36}\}$ with homogeneous ...
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An interpretation of Groebner basis in a special case

Let consider a free Lie algebra generated by $X$ with a set of relations $S$ such that the degree of leading monomial of relations in $S$ are greater than or equal to $2$. Let assume that we compute ...
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327 views

Minimal Gröbner bases have the same leading terms

Let $R=k[x_1,\ldots,x_n]$ for some field $k$. Let $I\subset R$ be an ideal, and let $G,G'$ be two minimal Gröbner bases for $I$. I want to show that $\text{LT}(G)=\text{LT}(G')$. That is, the set of ...
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61 views

Characterise the set of solutions of a nonlinear system of equations

Consider system of $8$ equations $$ \alpha^j(1-\alpha)^ip+(1-\alpha)^j \alpha^i (1-p)=q_{j,i} \hspace{1cm} \forall j\in \{0,1,...,7\}, i\in \{0,1,...,7\} \text{ s.t. } i+j=7 $$ where: $\{\alpha,p\}$ ...
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How to prove that if $F$ is a field then there exists only one monomial order in the polynomial ring $F[x]$?

My idea is to use proof by contradiction, that is to suppose there are two monomial orders $<, <'$ in $F[x]$ and somehow using the definition to conclude that. Where to start? I'm a little ...
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138 views

Alternative Gröbner Basis Packages

Hello helpful readers! The situation: I computed the solution space of $5$ variables with $4$ constraints as an expression of one free parameter. I did this using the groebner basis function in ...
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Prove: A basis $G$ is a Grobner basis of an ideal $\iff$ for every element $S$ in a homogeneous basis for the syzygies $S(G)$ we have $S.G \to_{G}0. $

A basis $G=(g_1,...,g_t)$ for an ideal I is a Grobner basis $\iff$ for every element $S=(h_1,...h_t)$ in a homogeneous basis for the syzygies $S(G)$ we have $S.G = \Sigma_{i=1}^{t} h_ig_i \to_{G}0. $ ...
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99 views

Two (different ?) definitions of a Gröbner-Basis

I have two slightly different definitions for Gröbner-Bases. 1.Definition from book Let $I$ be an ideal and $G=(g_1,\ldots,g_s)$ a basis for $I$. $G$ is called a Gröbner-Basis if $\langle LT(g_1),\...
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Finding Grobner basis using Buchberger's algorithm by Hand

I could use some help with this question. I have multiplied the first section by z and then subtracted the two polynomials from each other which has given me y=0, or x^2=z^2. Am I on the right track?...

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