All Questions
Tagged with polynomials computer-algebra
10 questions with no upvoted or accepted answers
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
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
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
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
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