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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Prove for the ideals $I,J$ that $J\nsubseteq I$

consider the ideals $I=\langle f_1,f_2\rangle$ and $J=\langle h_1,h_2\rangle$ in $\mathbb{Q}[x,y]$. I want to prove that $J\nsubseteq I$. I'm trying to do this by proving that every element from $J$ ...
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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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Solving polynomial systems on MATLAB

I'm working on a large polynomial system with 40+ variables and even more equations. I am aware that SINGULAR can ``solve'' it by finding its Groebner basis. However, I have to solve a sequence of ...
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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}\...
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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 ...
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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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How do i show that transitivity is required for a monomial order?

On Eisenbud - Commutative Algebra with a view Toward Algebraic geometry page 324 There is an example given why we must assume transitivity of a monomial order relation. this example, to me, seems ...
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Termination of Buchberger's algorithm for path algebras

I'm reading the paper Noncommutative Gröbner Bases, and Projective Resolutions by Edward L. Green, which presents a version of Buchberger's algorithm for path algebras. I'm trying to show that the ...
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Understanding Graver basis, Grobner basis, and Lawrence lifting

I'm trying to understand the relationship between Graver and Grobner basis, in particular how Graver basis can be computed via Grobner basis via Lawrence lifting. The key result appear to be Theorem 7....
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Differences in theory of Groebner bases when we work over finite fields.

I was having a look at the Buchberger algorithm as my work requires solving system of multivariate polynomial equations over finite fields. I am reading from here: Lecture 1 I came across the ...
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Suppose $G$ and $G'$ are grobner bases for the ideal $I$. Show that $\overline{f}^{G} = \overline{f}^{G'}$ for $f \in k[x_1, \cdots, x_n]$

Suppose $G$ and $G'$ are grobner bases for the ideal $I$. Show that $\overline{f}^{G} = \overline{f}^{G'}$ for $f \in k[x_1, \cdots, x_n]$ By the division algorithm and Proposition $1$, we may write $...
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Basic question about S-polynomial example

consider $f = xy + z^3$ and $z^2 -3z$ with Lex ordering (x>y>z). Then the S-polynomial of $f$ and $g$ is: $S(f,g) = xyz^2 + z^5 -xyz^2 - 3xyz = z^5 - 3xyz$. An S-polynomial is supposed to cancel ...
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Show that a sequence of polynomials cannot be a Gröbner basis wrt. any term ordering

Consider the ideal $I=\left\langle x^{2}+y^{2}, x^{3}+y^{3}\right\rangle \subseteq \mathbb{Q}[X, Y]$ and the basis $G=(x^2+y^2,x^3+y^3)$ with lexicographic ordering $x\geq y$. Consider two $S$-...
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Is the inverse of $(x+y, zw+z+w, y^2)\subset Q[x,y,z,w]$ under inclusion $g: Z[x,y,z,w]\to Q[x,y,zw]$ in $Z[x,y,z,w]$ still be $(x+y, zw+z+w, y^2)$?

In general, suppose $g: Z[x,y,z,w] \to Q[x,y,z,w]$ is the canonical inclusion, and $I=(f_1,...,f_n)\subset Q[x,y,z,w]$ be an ideal such that for each $f_i$, its coefficients are 1, and for each two ...
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Are there any examples of math olympiad problems that can be solved by modern math?

I am looking for word problems that can be tackled by subjects like category theory, commutative algebra, nonlinear algebra, algebraic geometry etc.
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Product of Grobner bases and product of the ideals generated by said Grobner bases

Suppose $G, G'$ are Grobner bases for the ideals $I, I' \subseteq F[x_1, \dots, x_n]$ respectively. Then is it the case that $GG'$ is a Grobner bases for the ideal $II'$? My intuition on this is no, ...
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"Mod" out symmetry in ideal for a Groebner basis calculation (using a quotient ring?)

Consider a set of polynomials $P$ in the polynomial ring $R$ of $n$ variables ($R = \mathbb{C}[x_1,...,x_n]$), and let $I$ be the ideal generated by the polynomials in $P$. I have an ideal which is ...
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Polynomial solutions to a polynomial using syzygies

$\newcommand\C{\mathbb C}\newcommand\syz{\operatorname{syz}} \newcommand\x{\mathbf x}\newcommand\a{\mathbf a}$Let $\a=a_1,\dots,a_m$ and $\x = x_1,\dots, x_n$ be indeterminates and consider a ...
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Gröbner bases method to solve system of polynomial equations

I am new to the theory of solving systems of polynomial equations using Gröbner bases and I am confused about the terminology. I will appreciate any help to clarify my confusion. Some sources are ...
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Finding $a+b+c+d$, where $ab+c+d=15$, $bc+d+a=24$, $cd+a+b=42$, $da+b+c=13$

Let $a,b,c,d \in \mathbb{R}$. Consider the following constraints: \begin{cases} ab+c+d=15 \\ bc+d+a=24 \\ cd+a+b=42 \\da+b+c=13 \end{cases} Calculate the value of $a+b+c+d$. It is easy to use the ...
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What's the proper way to take a Groebner basis with respect to a quotient polynomial ring?

Suppose I have the quotient ring $R=\mathbb{Q}[x,y]/I$ for some ideal $I$, and I want to find a Groebner basis for another ideal $J\subseteq R$. When computing the basis, does it make a difference if ...
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Use of Grobner Basis to Compute Relations with respect to Quotient Polynomial Ring

I think I might have phrased the question poorly, but I have a somewhat basic question about how to use Grobner bases in a particular example. Suppose I have some algebra $$R=\mathbb{C}[x_1,x_2^3,x_3^...
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How do I proof Groebner basis existence in $R[x_1, \dots, x_n]$?

In An Introduction To Groebner Bases from Loustaunau, it says that: We further note that the Noetherian property of the ring R, and hence of the ring $R[x_1, \dots, x_n]$, yields to the following ...
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Polynomials which are invariant to the cyclic permutation of variables

I'm trying to solve the following problem from this book. I can find the Gröbner basis of $J$ using Buchberger’s algorithm, and so I don't have any problem with the first part of this problem. But my ...
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Algorithm for expressing Gröbner basis in terms of ideal generators

Given a polynomial ring and an ideal $$A \supset I = (f_1, ..., f_m)$$ there are plenty of implementations of an algorithm (e.g. Buchberger's) that produces a Gröbner basis $$G = (g_1, ..., g_n)$$ and ...
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Given a set $G$, how do we show that it is a Gröbner basis of an ideal $I$?

I am using the sage-notation so it might be a bit untidy. I have been given a set $G$ and I want to show that it is a Gröbner basis of the ideal $I= \langle y_1+y_2x_1+y_3x_1^2 , y_1+y_2x_2+y_3x_2^2,...
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Write an ideal as an intersection of maximal ideals.

I'm trying to solve the following problem: Write $$ I = (x^2+2y^2-2,x^2+xy+y^2-2) \subset \mathbb{R}[x,y] $$ as an intersection of maximal ideals. I have used the Gröbner basis and I found that the ...
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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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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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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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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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2 votes
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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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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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2 votes
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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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3 votes
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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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