Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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.

321 questions
74
votes
8 answers

Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean algorithm and Gaussian elimination". I've tried to…
J. M. ain't a mathematician
  • 76,540
22
votes
4 answers

Applications of Gröbner bases

I would like to present an application of Gröbner bases. The audience is a class of first year graduate students who are taking first year algebra. Does anyone have suggestions on a specific application that the audience would appreciate?
Mykie
  • 7,245
17
votes
2 answers

Prove that determinant of a matrix (with polynomial entries) is non-zero

For $\mathbf x\in\mathbb (0,1)^n$ with $n>2$ and a positive integer $1\le k
Sergio Parreiras
  • 3,853
15
votes
3 answers

How do you solve a system of quadratic equations?

I was watching a video where a problem in Galois theory was posed such that it became necessary to tell if a certain element was a perfect square in a finite field extension of the rationals. By writing a general element in terms of its…
Tac-Tics
  • 2,353
11
votes
3 answers

Symmetric polynomials and the Newton identities

I want to write $P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$ in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the fundamental theorem of symmetric polynomials using…
Abelsh
  • 1,073
8
votes
1 answer

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 $J \cap \mathbb{Q}[y]$. Compute a generating set for $J \cap(y)$. Compute a…
Mathsstudent147
  • 1,573
7
votes
2 answers

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 Gröbner basis to get the…
Modal
  • 81
  • 1
7
votes
1 answer

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 I consider $J$ as a subset of $\mathbb{Q}[x,y]$,…
Calvin Godfrey
  • 1,574
7
votes
3 answers

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 have tried Let's denote the center of the circle…
minkbag
  • 554
7
votes
1 answer

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 $S(f_1,f_2)$. Most of the time $S(f_1,f_2)$ is an ugly…
Joachim
  • 2,663
7
votes
2 answers

Gröbner Basis and Division Algorithm

I recently read a lemma on a course in Commutative Algebra that states, If $G$ is a Gröbner Basis for an Ideal $I$ in $k[x_{1},...,x_{n}]$, then a polynomial $f$ belongs to $I$ if and only if $f$ on division by $G$ (we can do this using the…
user1314
  • 1,558
7
votes
4 answers

Calculating Gröbner basis for Sudoku

I'm trying to write a program that solves sudokus using a Gröbner basis. I introduced 81 variables $x_1$ to $x_{81}$, this is a linearisation of the sudoku board. The space of valid sudokus is defined by: for $i=1,\ldots,81$ : $F_i = (x_i - 1)(x_i -…
Ingdas
  • 211
6
votes
1 answer

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 lone monomial in $I$ also divides half of one of…
Trevor
  • 382
6
votes
1 answer

Largest subgroup in which a given polynomial is invariant.

I am trying to solve the following question; Given a polynomial $f\in \mathbb{C}[x_{1},x_{2},\ldots,x_{n}]$, find the largest subgroup $\Gamma\le GL(\mathbb{C}^{n})$ such that $f\in \mathbb{C}[x_{1},x_{2},\ldots,x_{n}]^{\Gamma}$. Here $f\in…
user551642
  • 704
6
votes
1 answer

How is Buchberger algorithm a generalization of the Euclid GCD algorithm?

It is said in many places (for example, on the Wikipedia article for Buchberger's algorithm) that Buchberger's algorithm to find Groebner basis is a generalization of Euclid's GCD algorithm. This is not obvious to me. Just think about two…
Herman Jaramillo
  • 3,280
1
2 3
…
21 22