All Questions
Tagged with groebner-bases polynomials
22 questions
0
votes
1
answer
80
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 $...
- 103
1
vote
0
answers
259
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 ...
- 21
3
votes
0
answers
114
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 ...
- 31
1
vote
1
answer
174
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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 ...
- 333
0
votes
0
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67
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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 $n\times m$ ...
- 344
0
votes
0
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55
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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 ...
- 1
1
vote
1
answer
326
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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 ...
- 1,150
1
vote
1
answer
108
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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 $\...
- 113
4
votes
0
answers
80
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 $k$. ...
- 476
2
votes
2
answers
3k
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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 ...
- 23
4
votes
1
answer
547
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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 ...
- 415
1
vote
0
answers
126
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, ...
- 415
13
votes
3
answers
949
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 $p$) ...
- 25.5k
1
vote
1
answer
470
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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}{{...
- 1,990
2
votes
3
answers
2k
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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 $f,g$.
...
- 1,990
12
votes
1
answer
2k
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Ways to show a system of polynomial equations has no solution
I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$:
$$
\begin{cases}
2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\
X_sX_{m-s}+(-1)^...
- 767
4
votes
0
answers
168
views
Bound for the height of equations defining the singular locus of a variety
Fix positive integers $m, n, d$.
In what follows, the height of an algebraic number will mean the absolute multiplicative height.
Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
- 1,500
7
votes
2
answers
872
views
Dimension of a homogeneous polynomial system
Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations:
\begin{cases}
&\sum\limits_{s=1}^{2t-1}z_sz_{2t-s}+\sum\limits_{s=2t+1}^{m-1}...
4
votes
0
answers
465
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Bounding the degrees in a Bézout relation for integer polynomials
Let $A$ and $B$ be two polynomials in $\mathbf Z[X]$ which generate $\mathbf Z[X]$, that is assume that there exist polynomials $U$ and $V$ in $\mathbf Z[X]$ such that
$$
A \cdot U + B \cdot V=1.
$$
...
- 2,521
5
votes
1
answer
3k
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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 ...
- 53
4
votes
2
answers
3k
views
Numerical solution for a system of multivariate polynomial equations
Hi all,
I have a system of 6th-order polynomial equations in 4 variables $q_1, q_2, q_3, q_4$ (i.e. polynomials with all the terms such as $q_1^6, q_2^6, q_2^4 q_3^2$):
$P_k(q_1, q_2, q_3, q_4) = 0$ ...
- 113
4
votes
4
answers
1k
views
Systems of polynomial equations
Hi all,
I'm an engineer assigned to determine some parameters of a manipulator (ie., calibration). It has a number of parameters, but after some manipulations of its dynamic equations, I can have the ...
- 41