Results for query "How to create skew polynomial in MAGMA"

Determining existence of p-adic point on a plane curve

Is there an implemented algorithm that will take a polynomial $f(x,y)\in\mathbb Z[x,y]$ and a prime $p$, and determine whether the equation $f(x,y)=0$ has a solution over $\mathbb Q_p$?My recollection is that Nils Bruin did this is in fair generality in Magma.

Finding relations between invariant polynomials

I already have the candidates for the generators (certain polynomials of $16$ variables) but I need to find the relations between them.My question is the following - is it possible to force Magma, Sage or other program to find these polynomial relations (hopefully the whole ideal of relations.

Algebraic integers in skew fields

Hi everyone,

let $D$ be a skew field, which is finite dimensional over its center $k$.Assume that $k$ is a number field, and let $\mathcal{O}_D$ be the set of elements $z\in D$ which are roots of a monic polynomial with coefficients in $\mathcal{O}_k$.

Global dimensions of non-commutative rings

Moreover, are there any standard way to compute $gl\dim(S)$ when $S$ is non-commutative?Then let $\sigma_1$ be the $k$-algebra automorphism of $R_1$ defined by $\sigma_1(x_1) = a_{21} x_1$, and let $R_2$ be the skew-polynomial ring
R_2 = R_1[x_2; \sigma_1].

Detecting if a polynomial is a Pfaffian

The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ is skew symmetric, or explicitly $${\rm pf}(A) = \frac{1}{2^n n!Let $F \in k[x_0, \ldots, x_n ]$ be a homogeneous polynomial of degree $d$.

Skew polynomial algebra

When I was a very little hare, a big grey wolf told me about the following skew polynomial algebra, which I never understood.My question is whether the following construction is a part of some bigger abstract construction and whether it is been written anywhere.

Square root of the determinant of AB+I where A, B are skew-symmetric

Imagine I have two skew-symmetric square matrices $A$, $B$.) Now I am interested in the square root of the determinant of $AB+I$, where $I$ is the identity matrix,

$$ x = \sqrt{ \det \left( AB + I \right) } $$

As quick inspection for small matrices suggests that this $x$ is a polynomial of the elements of $A$ and $B$, for example for $3 \times 3$ matrices we find

$$ x = 1 - a_{12} b_{12} - a_{13} b_{13} - a_{23} b_{23} $$

and I checked this analytically for matrices up to $6 \times 6$.

Almost skew polynomial ring an integral domain?

Since Ore extensions of domains are Ore extensions, this question can be reformulated as follows: Since the relation look almost like those of a skew polynomial ring, can this ring in fact be written as a skew polynomial ring?Consider the $k$-algebra $R=k\langle x_i, c_i\rangle_{i=1,\ldots,n}$, $k$ not of characteristic 2,
subject to the following relations:

$x_ix_j = x_jx_i$ for all $i, j$
$x_ic_i = -c_i x_i$
$x_ic_j = c_j x_i$ for all $i\neq j$
$c_ic_j = -c_j c_i$ for all $i\neq j$

I haven't found a counterexample yet.