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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.

303 questions
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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
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Uniqueness of reduced Groebner basis - proof

can you please help me understand something. I defined minimal GB like this: Let $G = \{g1, . . . , gs\}$ be a GB of an ideal $I ⊂ k[x1, . . . , xn]$. Then $G$ is a minimal GB if and only if for each $i = 1, . . . , s$, the polynomial $LC(gi) = 1$…
Petra
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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 leading terms. Can someone please clarify what this…
KZ-Spectra
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I want a famous application of comprehensive grobner basis w.r.t. lex order and w.r.t. some conditions on parametric coefficients.

Let $I$ be an parametric polynomial ideal in $K[a_1,\cdots,a_m][x_1,\cdots,x_n]$ where $a_1,\cdots,a_m$ is a sequence of parameters and $x_1,\cdots,x_n$ is a sequence of variables. Is there any famous example or application which in it we must…
mahdi dehghani
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How to prove using Groebner Bases that $x^2 +y^2 = −1, x^3 +y^3 = −1, x^5 +y^5 = −1$ is inconsistent in $\Bbb C\;^2$?

How can it be proved using Groebner Bases that the following system of equations is inconsistent in $\Bbb C\;^2$ ? $x^2 +y^2 = −1, x^3 +y^3 = −1, x^5 +y^5 = −1$
Alfonso_MA
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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},\dots,\overline{g_k})\subseteq…
sugyman
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How to show Buchbergers algorithm terminates in noncommutative cases

I formulated buchberger's Algorithm over certain types of Ore-Algebras, meaning, i have the case $A:=R[y_1,...,y_n]$ with $R$ being a commutative Ring and $y_1,...,y_n$ being noncommutative variables (regarding the Elements of $R$) with the property…
Kolja
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Groebner basis of a maximal ideal

is it a true for a maximal ideal $I=\langle x-a,\,y-b\rangle$ the vector space $\mathbb{C}[x,y]/I$ always has the dimension one? I thought we would have a Groebner basis $G$ of the same form as $I$ and $\mathbb{N}^2\setminus deg(G)=(0,0)$ therefore…
Alex
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find a basis and the dimensions of the solution space w

$$x+2y-2z+2s-t=0$$ $$x+2y-z+3s-2t=0$$ $$2x+4y-7z+s+t=0$$ I need to find the basis and dimensions. I'm not sure how to do it. The book I have doesn't have a very good example. I end up with: $$ \left( \begin{array}{ccc} 1 & 2 & 0 & 4 & -3 \\ 0 & 0…
kevorski
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Why are the number of Groebner bases for a set of polynomials sometimes greater than the number of polynomials

Its easy to see why a set of $n$ polynomials $\{p_1\cdots p_n\}$ may have fewer than $n$ Groebner bases. For example, if $p_n=p_1*p_2+p_3$ then $\{p_1\cdots p_{n-1}\}$ and $\{p_1\cdots p_n\}$ generate the same ideal. As a result I would expect it to…
Abijah
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