All Questions
Tagged with polynomials computer-algebra
32 questions
27
votes
5
answers
7k
views
Minimal polynomial of cos(π/n)
I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\...
- 1,491
21
votes
5
answers
3k
views
How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?
Example: How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this ...
- 24.9k
14
votes
2
answers
635
views
Tarski-Seidenberg for strict inequalities and bounded quantification
This theorem says that quantifiers over real variables can be eliminated from classical first order formulae built from equations and inequalities between polynomials with rational coefficients, ie in ...
- 8,481
11
votes
1
answer
475
views
Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials
Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...
- 63.1k
8
votes
3
answers
2k
views
Rational exponential expressions
Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...
- 2,283
6
votes
1
answer
338
views
Check irreducibility of an explicit polynomial, without computer
I have a polynomial of degree 8 in 6 variables given explicitly by
$$ (\sqrt{1+(x_1+x_2+x_3)^2+(y_1+y_2+y_3)^2}+\sqrt{1+x_1^2+y_1^2}+\sqrt{1+x_2^2+y_2^2}+\sqrt{1+x_3^2+y_3^2})\times\text{the other ...
- 5,169
5
votes
1
answer
1k
views
Is there an algorithm to decide if an ideal contains monomials?
Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one?
Gröbner bases come to ...
- 1,961
4
votes
1
answer
155
views
Is there a point in 6-dimensional space satisfying these polynomial inequalities?
I would like to know if there is a point $(a, b, p, q, x, y) \in [0,1]^6$ satisfying the following collection of inequalities.
$b \ge a$
$q \ge p$
$y \ge x$
$a \ge p \ge a^2$
$b \ge q \ge b^2$
$p \ge ...
- 543
4
votes
1
answer
334
views
GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
- 63.1k
4
votes
1
answer
206
views
Software computing dimension and degree
Assume a projective scheme $X_{k_1,\dots,k_r}\subset\mathbb{P}^n$ is given as the set of common solutions of homogeneous polynomials $F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$, where the $F_i$ ...
user114666
4
votes
1
answer
485
views
All rational periodic points
I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes ...
- 41
3
votes
2
answers
176
views
Algorithms (or packages) to find recurrence relations for given sequence of q-polynomials?
Assume we have sequence of polynomials : $P_i(q)$ - each term is polynomial in $q$. (With integer coefficients, but hopefully it is not important).
We expect that there exists recurrence relation a ...
- 24.9k
3
votes
1
answer
273
views
Solving polynomial inequalities -- efficient Positivstellensatz on a computer
I have about twenty five (multilinear) polynomials $f_1(\mathbf{x}), f_2(\mathbf{x}), \dots, f_{25}(\mathbf{x})$ all in fifteen variables and I would like to decide if there is a $\mathbf{y} \in [0,1]^...
- 543
3
votes
0
answers
150
views
For which $n$ is this ring an euclidean domain?
Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$.
Question: Is $A_n$ for all $n$ an euclidean domain? Is there a good choice for an euclidean function?
...
- 26.5k
3
votes
0
answers
98
views
Deterministic procedure to find irreducible polynomials
In $\Bbb F_q[x_1,\dots,x_n]$ given $d_1,\dots,d_n\in\Bbb N$ is there a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=\max_{i\in\{1,\dots,n\}}d_i$ and $d_i=deg(...
- 13.9k
2
votes
1
answer
590
views
A question on a Macaulay2 computation
I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables.
Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...
user125056
2
votes
3
answers
543
views
Useful software for variable elimination
I have three non-homogeneous trivariate polynomials in $\mathbb Z[x,y,z]$ and I want to eliminate the variables $y$ and $z$ to get a polynomial in $x$. The monomials of the polynomials are $\{1,x^4,x^...
- 13.9k
2
votes
0
answers
148
views
How to decide if an algebraic number is a root of a given polynomial?
Let $p$ be a polynomial with rational coefficients and $\alpha = \sqrt[n]{q}e^{i2k\pi/m}$ for some natural numbers $n,m,k$ and a rational number $q > 0$. Is there an effective algorithm for ...
- 147
2
votes
0
answers
78
views
Affine projection of polynomials for a given set of points
(Not sure this question fits here, I will remove it in case it doesn't)
Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a ...
- 53
1
vote
2
answers
134
views
Non-Uniform Root of Polynomial in Open Cube
I'm looking to find a root $(x_1,\dots,x_n)$ of a polynomial $p \in {\mathbb R}[x_1,\dots,x_n]$ such that $0 \leq x_i < 1$ for all $i$. Further, I know in advance that setting $x_1 = \cdots = x_n$ ...
- 432
1
vote
1
answer
174
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 ...
- 333
1
vote
1
answer
108
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 $\...
- 113
1
vote
1
answer
146
views
Omitting constraints of polynomial system
Let $n_1, n_2 \geq 1$ be known integer constants.
Suppose that we have the following system of $n$ polynomial inequalities
for which we know that there exists a feasible solution $(p_1, p_2) \in (0,1)...
- 175
1
vote
0
answers
108
views
Primitive element theorem for algebraic functions
Given a function $f(x) : \mathbb{R}^n \to \mathbb{R}$, we call it algebraic if it satisfies a polynomial equality $g(y, x) = 0$.
This is analogous to an algebraic number being the root of a univariate ...
- 326
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
1
vote
0
answers
62
views
Finding multivariate binomials with a common zero [closed]
I have a problem for which I have to find binomials over a multivariate polynomial Ring which all have a common zero.
Let $\mathbb{F}[x_1,\dots,x_n]$ be some multivariate polynomial ring over some ...
1
vote
0
answers
139
views
Polynomial systems and algebraic functions
An algebraic function $y(x)$ is defined as the solution of a polynomial equation of the form $p(x,y)=0$, that is one making the identity $p(x,y(x))=0$ true --- in either analytical or formal power ...
- 333
1
vote
0
answers
176
views
Coefficient perturbations of polynomials with real roots only
Let
\begin{align}
P(x) &= x^n+a_{n-1} x^{n-1} +\ldots+a_0 = \prod_{i=1}^n (x-p_i)\\
Q(x) &= x^n + b_{n-1} x^{n-1} + \ldots +b_0 = \prod_{i=1}^n (x-q_i)\\
p_i, q_i& \in \mathbb{R},\ 0<...
- 175
1
vote
0
answers
280
views
Algebraic independence criterion
Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
- 11
0
votes
1
answer
509
views
Polynomial of degree N with integer coefficient for a given root.
Is it possible to construct a polynomial of degree N, with all of them as integer coefficient have a root as the given value. ...
- 23
0
votes
1
answer
73
views
Relations between non-negativity of multivariate polynomials and SOS over gradient ideal
We know there is a necessary condition for the non-negativity of multivariate polynomials in the paper "Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational ...
- 59
-5
votes
1
answer
79
views
Application of Resultant in Computer Algebra [closed]
Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
