# 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.

263 questions
Filter by
Sorted by
Tagged with
40 views

### 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 $J$ ...
1 vote
28 views

### 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 ...
1 vote
236 views

1 vote
48 views

### 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 ...
1 vote
77 views

### 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 ...
1 vote
88 views

### 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 ...
100 views

95 views

### 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 ...
48 views

### 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? 1 vote
65 views

### 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 ...
183 views

### 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 ... 302 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 ... 213 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 ... 159 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 ...
1 vote
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 $k[x_{1}... 1 vote 1 answer 62 views ### Does this square polynomial system have a finite number of zeros? Wish to know if this square polynomial system has a finite number of zeros:$\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}+\... 0 votes 1 answer 117 views ### 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: <... 0 votes 1 answer 45 views ### 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) ... 2 votes 1 answer 49 views ### Invert "matrix of polynomials" for a change of presentation of an Ideal If I start with a polynomial ideal generated by a set of polynomialsf_i$, which is not a standard basis for the ideal, and then I obtain a standard basis for the ideal to be a set of polynomials$...
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 ...