# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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?...