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Solving sparse bilinear systems with a relatively large number of variables

I'm trying to solve a bilinear system of equations over a finite field. (More specifically: I'm trying to find a single solution, if one exists.) The system consists of equations of the form $$y^T A_i ...
Sic Vis's user avatar
  • 101
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0 answers
59 views

The theory of Groebner bases in Jordan case

There are many papers regarding spreading the theory of Groebner-Shirshov bases from Lie algebras to other nonassociative algebras. Also, it has been studied for associative algebras with operators ...
user118746's user avatar
2 votes
1 answer
113 views

Efficiency of Groebner basis for constraints of the form $(a_i x_i+b_i)(a_j x_j+b_j)$

This is based on numerical experiments in sage. Let $K$ be a ring and define the ideal where each polynomial is of the form $(a_i x_i+b_i)(a_j x_j+b_j)$ for constant $a_i,b_i,a_j,b_j$. Q1 Is it true ...
joro's user avatar
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Extracting implications in polynomial constraint system from Groebner basis

Given a Groebner basis for a system of polynomial constraints over $\mathbb{Q}$, are there any known methods for extracting the low degree factorable polynomials in the ideal generated by that basis? ...
PPenguin's user avatar
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1 vote
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Gorenstein property from initial ideal

My question is: If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...
Chess's user avatar
  • 13
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1 answer
78 views

Ideal membership and change of fields

Let $R=k[x_1,...,x_n]$ be a polynomial ring over a field. Let $f$ be a homogeneous polynomial and $I=(f_1,...,f_m)$ a homogeneous ideal. With Macaulay2, one can compute the Groebner basis of $I$ when $...
T C's user avatar
  • 103
2 votes
0 answers
82 views

Gröbner implicitization with relationships between the variables

I have the following parametric equations, where cost$=\cos t$, cos2t$=\cos 2t$, and $A^2+B^2=1$: ...
Stéphane Laurent's user avatar
3 votes
1 answer
244 views

Resultants and elimination theory

Consider an ideal $I = \langle f_1,\dotsc,f_n\rangle$ in the ring $k[x_1,\dotsc,x_m]$. Define the $i$-th elimination ideal to be $I_i = I \cap k[x_{i+1},\dotsc,x_m]$. For any two polynomials $f$ and $...
giulio bullsaver's user avatar
0 votes
1 answer
202 views

Are zero dimensional ideals radical?

I have a question about Theorem 3.7.25. of Computational commutative algebra I by M. Kreuzer and L. Robbiano. Let $K$ be a perfect field, $I \subseteq K[x_1, \ldots, x_n]$, be a zero dimensional ...
Mairon's user avatar
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7 votes
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173 views

How can Gröbner bases be generalized to differential algebra?

I'm aware of the Rosenfeld-Gröbner algorithm "for computing a regular decomposition of a radical differential ideal generated by a set of polynomial differential equations, ordinary or with ...
Brent Baccala's user avatar
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1 answer
183 views

Compatibility conditions for quadratic equations

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$: \begin{eqnarray} 0 &= A_1x^2 + B_1x + C_1 \\ &...
Michael Schindler's user avatar
5 votes
3 answers
1k views

How to speed up the process for calculating the Groebner basis?

I am currently trying to get the Groebner basis for 9 equations with 12 variables: $ a_1^2+b_1^2+c_1^2+d_1^2-48.73=0\\ a_2^2+b_2^2+c_2^2+d_2^2-50.53=0\\ a_3^2+b_3^2+c_3^2+d_3^2-40.69=0\\ a_1a_2+b_1b_2+...
Gabriel's user avatar
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3 votes
1 answer
334 views

Groebner basis with parameters

I need to compute a Groebner basis of a polynomial system with parameters. The only recent results I found is Groebner cover: https://www.sciencedirect.com/science/article/pii/S0747717110000970 Are ...
Dragomir's user avatar
0 votes
1 answer
173 views

Grobner basis of a submodule of a free module over polynomial ring

Let $A=\mathbb{Q} [x_1,\dots,x_n]$ be the ring of polynomials with rational coefficients. Let $T$ be an $m\times n$ matrix with entries from $A$. Consider it as a morphism of $A$-modules $T\colon A^n\...
asv's user avatar
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2 votes
1 answer
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Library/Database of parametric polynomial systems

Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters? I need some real examples to test my ...
Ayoola Jinadu's user avatar
4 votes
0 answers
115 views

Recommendations for distributed calculations of Groebner Bases

There are many computer algebra systems available which can compute a Groebner basis, including: Mathematica Singular Macaulay2 Magma Maple CoCoA However (please correct me if I've missed something) ...
JoggingGrad's user avatar
2 votes
1 answer
180 views

Techniques for estimating dimension of algebraic variety without calculating Groebner basis of polynomial constraints?

I have a system of multilinear polynomial constraints that I believe has no solution. Unfortunately, the number of variables and number of constraints is too large to afford direct computation of a ...
JoggingGrad's user avatar
1 vote
0 answers
229 views

Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA

I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently. So far, I only found MAGMA with its ...
ArminJR's user avatar
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4 votes
0 answers
195 views

Buchberger's criterion for Gröbner bases in $k$-algebras with multiplicative basis and admissible order

Let $R$ be an associative $k$-algebra with multiplicative basis $\mathcal B$ with an admissible order on $\mathcal B$. Let $G \subseteq R$ be a subset. A multiplicative basis $\mathcal B$ means that $...
Berber's user avatar
  • 141
2 votes
1 answer
591 views

Efficiently computing Gröbner basis to prove no solution to polynomial constraints

In a similar vein to these now quite old questions on advice for calculating a Gröbner basis: Fast computation of a Groebner basis. What is possible? What is the state of art in Groebner bases I am ...
RedCurry's user avatar
3 votes
0 answers
95 views

Methods for multivariate polynomial equations over large finite fields

I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given ...
sugyman's user avatar
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1 vote
1 answer
165 views

Sufficient syntactic conditions for zero-dimensionality of polynomial systems

Consider a system $S$ of polynomial equations, $p_1=0,...,p_m=0$, for $p_i\in K[x_1,...,x_n]$, for a field $K$: the system $S$ is zero-dimensional if it has finitely many solutions. It is well-known ...
Michele's user avatar
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0 votes
0 answers
66 views

The minimum number of polynomial equations the components of linearly dependent vectors must satisfy

Context: Consider $m<n$ vectors $v_1,\dotsc,v_m\in\mathbb{C}^n$ with complex components. We can study if they are linearly dependent by constructing the following matrices. First the $n\times m$ ...
FeedbackLooper's user avatar
0 votes
0 answers
54 views

Dependence of the complexity of solving polynomial sytems on the multidegree

Let $f_1,\ldots,f_n\in \mathbb{Q}[X_1,\ldots,X_n]$ be a system of $n$ polynomials in $n$ indeterminant, which only has finitely many solutions. Supose that the each of the variables $X_i$ appears at ...
Ben's user avatar
  • 1
0 votes
0 answers
70 views

Low rank approximation

Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...
anjan samanta's user avatar
10 votes
0 answers
533 views

How general are Gröbner degenerations?

While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ...
Igor Makhlin's user avatar
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2 votes
0 answers
68 views

grobner basis of an ideal dependent on some parameter

Suppose $I = \langle f_1, ... , f_l \rangle$ is an ideal generated by polynomials $f \in k[x_1,\dots,x_n]$, where $k$ is a field of rational functions in some parameters $s_1,\dots,s_m$. What are the ...
giulio bullsaver's user avatar
1 vote
0 answers
123 views

Groebner basis of a toric ideal

I know about toric ideals that it is a sort of binomial ideal i.e. generated by $x^u - x^v$, where $Au = Av $ ( A is the associated matrix). So by finding all integer solutions of $AX = 0$, can we ...
anjan samanta's user avatar
0 votes
0 answers
119 views

Is the Hilbert series of an ideal related to the Hilbert series of its homogenization?

Suppose we have a field $k$ of characteristic 0, let $I$ be an ideal of $R=k[x_1,...,x_n]$, and let $H$ be the homogenization of $I$ in $S=R[z]$. Is there any relationship between the Hilbert series ...
Stephen McKean's user avatar
1 vote
1 answer
222 views

Gröbner basis via integer programming

I have studied some papers related to solving integer programs via Gröbner bases. I wonder if the other way is possible or not — i.e., given any ideal, can we find the Gröbner basis by translating ...
anjan samanta's user avatar
1 vote
0 answers
107 views

How to obtain a linear basis from a Groebner basis?

Let $A$ be a finite dimensional algebra generated by $x_1, \ldots, x_n$ subject to certain relations $I_1, \ldots, I_m$. Could we obtained a linear basis $B$ consisting of monomials in $x_1, \ldots, ...
Jianrong Li's user avatar
  • 6,141
6 votes
1 answer
254 views

Groebner Bases for submodule over polynomial ring with integer coefficients

It is well-known that there exist Groebner bases for ideals in polynomial ring $\mathbb Q[x]$ which can be found algorithmically Moreover, I don't think it is hard to show that there exist Groebner ...
Wenhao Wang's user avatar
8 votes
1 answer
640 views

Given a zero-dimensional ideal $(f_1,...,f_n)$, is the ideal $(f_1-\alpha_1,...,f_n-\alpha_n)$ also zero-dimensional?

Suppose you have a zero-dimensional ideal $I=(f_1,...,f_n)$ in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$, so that $\dim_k(R/I)<\infty$. Take indeterminates $\alpha_1,...,\alpha_n$ and ...
Stephen McKean's user avatar
2 votes
0 answers
224 views

Computing the codimension of the variety defined by a system of quadratic forms

Suppose I have an $m \times n$ matrix $L$, where $m \leq n$ and each entry is $L_{i,j}(x_1, ..., x_s)$ which is a linear form over $\mathbb{C}$. Let $\mathbf{y} = (y_1, \ldots, y_n)$. Let us consider ...
Johnny T.'s user avatar
  • 3,605
4 votes
1 answer
349 views

Can a minimal generating set for an ideal always be made into a Groebner basis?

Let $I\subseteq k[x_0,\ldots,x_n]$ be an ideal, generated by some polynomials $F_1,\ldots,F_r$, all homogeneous and of the same degree. Suppose $r$ is the smallest number of generators that will ...
catfish's user avatar
  • 153
5 votes
1 answer
243 views

Infinitely many initial ideals for non-Artinian monomial orders?

Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...
Igor Makhlin's user avatar
  • 3,503
1 vote
1 answer
308 views

System of polynomial equations with a known root

I have 5 polynomial equations for 5 variables and I know that the set of roots is finite. All coefficients are integers. Ultimately I'd like to find all roots but finding the Groebner basis is ...
Fetchinson0234's user avatar
12 votes
2 answers
876 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< ...
Huy Dang's user avatar
  • 245
1 vote
1 answer
107 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 $\...
Pjotr5's user avatar
  • 113
3 votes
1 answer
167 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 $\...
Maxim's user avatar
  • 424
4 votes
0 answers
95 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 ...
John Palmieri's user avatar
1 vote
0 answers
35 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 $<_{rev}$ ...
Problemsolving's user avatar
4 votes
0 answers
100 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 $$ ...
Jürgen Böhm's user avatar
2 votes
1 answer
267 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 ...
W. Cadegan-Schlieper's user avatar
3 votes
0 answers
86 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 ...
Adrien's user avatar
  • 8,454
3 votes
1 answer
210 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 ...
Soutrik's user avatar
  • 155
3 votes
1 answer
183 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 ...
Alejandro Tiraboschi's user avatar
5 votes
2 answers
920 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#...
mayer_vietoris's user avatar
2 votes
1 answer
523 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 ...
teller's user avatar
  • 337
4 votes
0 answers
79 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 $k$. ...
Michiel Kosters's user avatar