Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

The tag has no usage guidance.

12
votes
2answers
723 views

Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$

This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
1
vote
1answer
74 views

Finding a characteristic for which the zero-locus of an ideal is not empty

I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\...
3
votes
1answer
138 views

Solving system of multivariable algebraic equations over $\mathbb Q$ by reducing over $\mathbb F_p$

I try to solve the finite system of multivariable algebraic equations with coefficients from $\mathbb Q$. It would be sufficient for me to prove that there is only finite number of solutions over $\...
4
votes
0answers
50 views

Compute equalizer of maps of polynomial rings, perhaps using Gröbner bases

Suppose that $k$ is a field and I have two ring homomorphisms $$ \phi, \psi :k[x_1,...,x_m] \to k[y_1,...,y_n]. $$ How can I use Gröbner bases (or other computational tools) to compute the subring of ...
1
vote
0answers
26 views

Grobner basis of the toric ideal $I_{A_P}$ with respect to $<_{rev}$ consists of those binomials $t_αt_β − t_{α\cap β} t_{α\cup β}$

I try to understand the proof of the Theorem. 10.1.3.(page 185) from ''Monomial Ideals'' by Herzog & Hibi. The reduced Grobner basis of the toric ideal $I_{A_P}$ with respect to $<_{rev}$ ...
4
votes
0answers
68 views

Gröbner bases of resultants and their monomial ideals

$\newcommand{QQ}{\mathbb{Q}}$ Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial $$ f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 - i} $$ Now let $$ ...
2
votes
1answer
140 views

Memory usage of Gröbner basis computation

I've been calculating some Gröbner bases in preparation for finding non-commutative Hilbert series (and, once I recreate that, characters of group actions). Specifically, I've been using the ...
3
votes
0answers
61 views

$\mathbb Z$-torsion for some quadratically presented Lie rings

$\newcommand{\Z}{\mathbb{Z}}$ I asked this question on MSE but no answer so far, so I'm also asking it here. Let $L$ be a Lie ring (a Lie algebra over $\Z$) with generators $x_1,\dots,x_n$ and ...
2
votes
1answer
153 views

Geometric significance of Anick's resolution

Given an augmented graded associative $K$ -algebra $A$, we can construct a free resolution of $K$ given by $K_A$ modules which gives nice combinatorial informations about the homology classes of the ...
1
vote
0answers
63 views

Size of the Groebner basis and change of coordinates

Given an ideal $I$ of $k[x,y]$, the size of the Groebner basis depends on the monomial ordering. For example, if $$ I = \langle x^3y^4 , x^2 + y^2 \rangle, $$ then the Groebner basis with the ...
5
votes
2answers
443 views

Automatic proof in Euclidean Geometry using Theory of Groebner Bases

I've done the same question in math.stackexchange here "https://math.stackexchange.com/questions/1938261/automatic-proof-in-euclidean-geometry-using-theory-of-groebner-bases?noredirect=1#...
2
votes
1answer
248 views

Integer programming and Groebner basis

I enjoyed reading different papers about using Groebner basis to solve integer programming. Is there any literature about the complexity and/or comparison with other (more classical) methods like ...
4
votes
0answers
65 views

Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let $k$ be a field and let $R=k[X_1,...,X_n]$ be a polynomial ring. Let $F \subset R$ be a finite subset generating a radical ideal $I$ with precisely $e$ solutions over an algebraic closure of $k$. ...
5
votes
0answers
201 views

Weyl algebra acting on a polynomial ring

Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots, x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl algebra. As usual $W$ ...
1
vote
0answers
54 views

Computing intersection of Weyl algebra ideal with certain subring

Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n] $ be the nth Weyl algebra over the characteristic zero field $k $. Set $\theta_i=x_i\partial_i $. Let $I $ be a left ideal in $D $. Is there a ...
5
votes
1answer
155 views

Groebner bases for differential operators with field coefficients (reference request)

Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements ...
5
votes
0answers
107 views

Computations in Weyl algebra with rational function coefficients

I am looking for a software to perform calculations with modules over the algebra $R_n=\mathbb{C}(x_1\ldots x_n)\langle \partial_1\ldots\partial_n\rangle$ of differential operators with rational ...
1
vote
2answers
1k views

Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...
1
vote
1answer
446 views

Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...
3
votes
1answer
239 views

How can I include irreducibility in a Groebner basis calculation?

I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques. For example, consider the equation $qn = mf$, where each of the variables ...
2
votes
0answers
83 views

Polynomial constraints triggered by irreducibility [closed]

I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic: $$af^2 + bf + c = 0$$ If we're working in a ring, ...
6
votes
2answers
371 views

Free $k[x_1, \dots, x_n]^{S_n}$-module?

Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} x_n^{...
4
votes
1answer
164 views

Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials \begin{align} & p_1(x_1,x_2,\ldots,x_n)=0, \\ & p_2(x_1,x_2,\ldots,x_n)=0, \\ & \vdots \\ & p_m(x_1,x_2, \...
12
votes
3answers
551 views

Polynomials vanishing modulo some integer $n$

It is well-known that a polynomial $q \in \mathbb Z[t]$ vanishes modulo $p$ only if it lies in the ideal $J_p$ generated by $p$ and $t^p-t$. This means that either the degree is large (at least $p$) ...
15
votes
2answers
533 views

From polynomial ideal over $\mathbb{Q}$ to polynomial ideal over $\mathbb{Z}$

Is there an algorithm to compute, given a polynomial ideal $I\subset \mathbb{Q}[x_1,\dotsc,x_n]$, the ideal $I\cap \mathbb{Z}[x_1,\dotsc,x_n]$ in $\mathbb{Z}[x_1,\dotsc,x_n]$? The input and output ...
2
votes
2answers
174 views

Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...
1
vote
1answer
365 views

cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian. Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where $f_k=\sum_{i=0}^{[k/2]}(-1)^{k-i}{{...
2
votes
3answers
650 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
0
votes
0answers
97 views

Maximal elements for ideals and subrings ordered by inclusion with fixed number of minimal generating polynomials

Let $R=\mathbb{R}[X_1,\dots,X_n]$, and $$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$ ...
1
vote
1answer
365 views

Is there a Gröbner basis analogue that exists for vector spaces?

Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$. ...
11
votes
1answer
1k views

Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$: $$ \begin{cases} 2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\ X_sX_{m-s}+(-1)^...
11
votes
4answers
3k views

Fast computation of a Groebner basis - What is Possible

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
4
votes
0answers
150 views

Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$. In what follows, the height of an algebraic number will mean the absolute multiplicative height. Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
2
votes
1answer
62 views

Degenerations and spanning monomials

Let $R = \mathbb{C}[x_1,…,x_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear ...
0
votes
2answers
279 views

Computing toric ideals via saturation and Groebner bases of toric ideals

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer. Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
7
votes
2answers
645 views

Dimension of a homogeneous polynomial system

Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations: \begin{cases} &\sum\limits_{s=1}^{2t-1}z_sz_{2t-s}+\sum\limits_{s=2t+1}^{m-1}...
4
votes
1answer
134 views

Algorithm to detect if an element of a (commutative) ring is in a subring?

For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring. Is there ...
-1
votes
1answer
97 views

How to recover the ideal from grobner basis of kernel of ann(x)

M -> ann(x) i can find the grobner basis of kernel of ann(x) and need the final step to recover this basis to ideal as i know, eliminate is not for all cases, what is the general practice to treat ...
0
votes
0answers
269 views

Can we find a Groebner Basis?

I would like to ask the following. Given only the leading terms of an ideal $I$, namely the set $LT(I)$, is it possible to find a Groebner Basis of $I$? If not always, then when is it possible? We ...
6
votes
3answers
498 views

Groebner bases for power series rings (reference request)

Hello, Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals. Thank you in ...
8
votes
2answers
343 views

Solving the Field Membership Problem using Grobner Bases

Is there an easy way to determine whether a set of elements in a field generates the whole field or only a subfield? Specifically, I have a subfield of $k(x,y)$ described in terms of a canonical set ...
6
votes
0answers
275 views

Testing isomorphism of finitely generated algebras

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and $C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated $\...
6
votes
1answer
498 views

Reasonable implementation of finding Gröbner bases over non-field coefficient rings

Gröbner bases are usually considered in the ring of polynomials over a field. However, there are useful definitions and algorithms for Gröbner bases over other coefficient rings; see, for instance, ...
14
votes
1answer
2k views

Gröbner basis for Sudoku

I'm trying to write a program that solves Sudokus using Gröbner bases. I introduced $81$ variables, $x_1$ to $x_{81}$, this is a linearisation of the Sudoku board. The space of valid Sudokus is ...
10
votes
2answers
367 views

Monomial orderings in noncommutative setting

An ordering of monomials in the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ is called a monomial ordering (EDIT: it seems that an equally common term used in this context is "term ...
4
votes
0answers
282 views

Bounding the degrees in a Bézout relation for integer polynomials

Let $A$ and $B$ be two polynomials in $\mathbf Z[X]$ which generate $\mathbf Z[X]$, that is assume that there exist polynomials $U$ and $V$ in $\mathbf Z[X]$ such that $$ A \cdot U + B \cdot V=1. $$ ...
2
votes
1answer
248 views

What does the d-slice of a weighted polynomial algebra look like?

This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves. ...
7
votes
1answer
452 views

Can we make Buchberger's algorithm faster for a given ideal if we are allowed to vary the monomial order?

Suppose we have a finite set of generators for an ideal $I \subset R := \Bbbk[x_1,\dotsc, x_n]$, where $\Bbbk$ is a field. If we choose a monomial ordering, then Buchberger's algorithm allows us to ...
3
votes
1answer
2k views

Existence of a real-valued solution to system of multivariate polynomial equations

Given a system of multivariate, polynomial equations, is there a way to determine if it has a solution in a given field (for instance the set of all reals). I don't care what the solution is, I just ...
8
votes
1answer
724 views

PBW theorem over a Q-algebra, without freeness or flatness

Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...